X-ray synchrotron radiation studies of actinide materials

By reviewing a selection of X-ray diffraction (XRD), resonant X-ray scattering (RXS), X-ray magnetic circular dichroism (XMCD), resonant and non-resonant inelastic scattering (RIXS, NIXS), and dispersive inelastic scattering (IXS) experiments, the potential of synchrotron radiation techniques in studying lattice and electronic structure, hybridization effects, multipolar order and lattice dynamics in actinide materials is demonstrated.

1. Introduction

Actinides are the heaviest chemical elements available on a macroscopic scale. The complexity of their electronic structure often produces exotic physical properties, such as the unconventional superconductivity of PuCoGa₅ and the hidden-order phases found in NpO₂ and URu₂Si₂ (Sarrao et al., 2002; Mydosh & Oppeneer, 2011; Moore & van der Laan, 2009; Santini et al., 2009). The richness of actinide physics has multiple origins. Intra-atomic Coulomb correlations compete with spin–orbit, exchange, and crystal field interactions on a comparable energy scale; scalar relativistic effects influence the spatial extension of the 5f electron shell; hybridization between 5f and conduction electron states is strong, attributing to the 5f electrons a dual localized-itinerant character.

X-ray synchrotron radiation techniques provide powerful tools to unravel the complexity of actinide materials. These elements- and shell-specific techniques probe spatial and temporal fluctuations of structural and electronic degrees of freedom, allowing one to observe hidden-order parameters and characterize elementary excitations with high sensitivity and resolution. Contrary to neutron scattering, synchrotron radiation experiments require samples on the microgram scale. This is important for actinides, as large quantities are difficult to manipulate and large single crystals are rare.

A central feature of the research on actinides can be illustrated by Fig. 1 and the atomic volume as a function of electron count across the 3d, 4f, and 5f series of elements. For the 3d elements, additional 3d electrons result in a contraction of the atomic volume as each additional electron adds to the cohesion of the element, resulting in a smaller atomic volume. In the 4f (rare-earth) series, apart from the two divalent elements Eu and Yb, the atomic volume is practically constant across the series. This is because the 4f electrons are spatially located close to the nucleus and are not involved in the bonding. However, for the 5f (actinide) series both behaviors are observed: an initial drop in the volume up to α-Pu, suggesting the 5f states are contributing to the bonding and are therefore itinerant for the light actinides, and then a strong expansion for other phases of Pu and through to heavier actinide elements, hence suggesting a localization of the 5f states from Am onwards.

Figure 1 Atomic volume of the transition (3d), rare-earth (4f), and actinide (5f) elements as a function of electron count. Note the unusual shape of the curve for the actinide series. In the case of Eu and Yb the large expansion is due to these elements being stable in the divalent state. All other rare-earth elements have the trivalent ground state.

Actinide elements, unlike most metals, crystallize in open-packed, low-symmetry structures. The behavior of elemental plutonium, exhibiting six ambient pressure allotropes, is exemplary (Zachariasen & Ellinger, 1955, 1957, 1963a,b; Lashley & Lawson, 2019). Its large volume expansion is unique in the actinides, and is particularly dramatic. As Pu is heated, its density decreases by about 20% from the room-temperature α-Pu form (simple monoclinic, space group P2₁/m) until reaching the high-symmetry face-centered cubic δ phase (), stable between 315 and 452°C. Then, plutonium contracts with a density increase of ∼4% while transforming to the body-centered cubic ε phase that eventually contracts on melting at 639.4°C. One obvious experiment is to compress Am and heavier actinides and see whether the resulting compressed elements have the open crystal structures found for the more itinerant lighter actinide elements.

2. X-ray diffraction

X-ray diffraction is certainly the technique of choice for determining crystallographic structures. Most of the ordinary crystallography is, however, carried out with conventional X-ray sources, and this is particularly true for actinides due to safety requirements. Synchrotron radiation facilities become necessary for high resolution or high pressure studies. The latter, in particular, have played an important role in the development of first-principles electronic structure calculations methods.

Electrons in the 5f shell are prone to an instability between localized and magnetic states to itinerant and bonding ones. Although metallic, elemental light actinides exhibit low-symmetry open-packed structures, which is unusual for simple metals. Moreover, a pressure of the order of a few GPa is sufficient to destabilize their crystallographic structure and induce a sequence of phase transformations. In some cases, the structural changes are accompanied by volume collapses indicating a stepwise delocalization of 5f electrons and their increasing contribution to the crystal cohesion energy.

Some examples are shown in Fig. 2 showing the equation of states for α-U, Am, and Cm as determined by high-pressure X-ray diffraction. Under compression, uranium preserves its room-temperature-stable orthorhombic Cmcm form up to at least 100 GPa. A different behavior is observed for americium where the normal pressure double-hexagonal close-packed (P6₃/mmc) structure transforms at 6.1 GPa to a face centered cubic () lattice. At higher pressures, two lower symmetry structures appear, a face-centered orthorhombic Am III (Fddd) and a primitive orthorhombic structure, Am IV (Pnma). In the same pressure range, up to 100 GPa, curium exhibits five phases, from the double-hexagonal close-packed (d.h.c.p.) form of Cm I (P6₃/mmc) to the ortho­rhombic (Pnma) structure of Cm V. Of particular interest is the formation of the monoclinic structure with the space group C2/c between about 37 and 56 GPa. Calculations based on the full potential linear muffin-tin orbital (FPLMTO) method suggest that its stabilization is driven by the magnetic correlation energy (Heathman et al., 2005). The collapse of Am and Cm from simple to complex structures under pressure shows, in keeping with Fig. 1, that the 5f electrons have transformed from localized to itinerant under pressure.

Figure 2 Pressure dependence of the normalized unit cell volume, V(p)/V0, for α-uranium (Le Bihan et al., 2003), americium (Lindbaum et al., 2001), and curium (Heathman et al., 2005). The inset shows a typical diamond anvil cell used for angle-dispersive, high-pressure X-ray diffraction experiments. Adapted from Heathman et al. (2005). Reprinted with permission from AAAS.

Beyond Cm, the available quantities become very small, but recent work has shown that the concept of de-localization under pressure may be too simple a picture. Examples are pressure work on Cf (Heathman et al., 2013) and spectroscopy work on Es (Carter et al., 2021), both using synchrotron radiation techniques.

Reproducing from first-principles electronic structure calculations the observed sequence of lattice geometry and the associated evolution of physical properties in actinide elements and compounds is challenging. Simple approximations of the density functional theory, in fact, are not adequate for actinide materials, because of the strong intra-atomic correlations, the importance of scalar (first-order kinetic energy correction due to the mass variation) and non-scalar (spin–orbit coupling) relativistic effects, and the extent of the hybridization between 5f and conduction electrons. In many cases a model that produces the correct crystal structure at a certain atomic volume fails to describe the electronic structure near the Fermi level, and does not reproduce the correct magnetic state. This is the case, for instance, of conventional density functional theory (DFT) calculations in the local-spin density or generalized-gradient approximation (LDA/GGA) applied to δ-Pu (Lashley et al., 2005; Joyce & Lander, 2019), or of static mean field correlated band theory calculations making use of different flavors of the LDA/GGA plus Coulomb's U (LDA + U) method, falling short in describing the itinerant-to-localized crossover of the 5f manifold in PuCoGa₅ (Daghero et al., 2012; Shick et al., 2013). During the last two decades, experiments such as those reported in Fig. 2 have, therefore, stimulated the development of increasingly sophisticated theoretical models that have now reached predicting capabilities close to a material-by-design level also for compounds in the 5f block (Shim et al., 2007; Suzuki et al., 2010; Pezzoli et al., 2011; Dudarev et al., 2019; Pourovskii & Khmelevskyi, 2021; Shick et al., 2021).

As an example of high resolution XRD study, we show in Fig. 3 the results obtained for the PuCoGa₅ unconventional superconductor (Eloirdi et al., 2017). The experiment was performed at the ESRF ID22 beamline, affording a resolution Δd/d = 10⁻⁶ at λ = 0.354155 Å. Data were collected on a 4.6 mg sample, obtained by crushing a single crystal grown from metallic plutonium (99.932 wt% ²⁴²Pu), put inside a hermetic holder providing four levels of containment. The absence of visible splitting or broadening of the diffraction peaks, as seen in the inset of Fig. 3 for the (2 2 0) Bragg reflection, indicates that the tetragonal symmetry is preserved in the superconducting phase. The temperature dependence of the refined lattice parameters shows that the thermal expansion is isotropic above ∼150 K. At lower temperatures, the c/a ratio increases with decreasing T. In the same temperature range, Ramshaw et al. (2015) observe a softening of the bulk modulus that they attribute to the development of in-plane hybridization between conduction electron and Pu 5f.

Figure 3 X-ray diffraction pattern recorded for PuCoGa5 at 5 K. The inset on the left shows the tetragonal crystallographic unit cell, with Pu and Co atoms represented by blue and red spheres, respectively, whilst the two inequivalent Ga atoms are shown by orange and yellow spheres. The inset on the right shows the (2 2 0) Bragg peak measured at 5 K (black circles) and 21 K (red circles). Reprinted with permission from Eloirdi et al. (2017). Copyright (2017) by the American Physical Society.

Below T_c = 18.5 K, the critical temperature to the superconducting state in PuCoGa₅, the expansion of the unit cell volume deviates from the predictions of a simple one-phonon Grüneisen–Einstein model (Fig. 4). The shrinking of the cell volume is similar to the one observed for the CeRu₂Si₂ Kondo system (Hiranaka et al., 2013) and could suggest the occurrence of critical valence fluctuations at T_c (Miyake & Watanabe, 2014), where the volume thermal expansion coefficient α_V has a jump larger by a factor of ∼20 than the value predicted by the Ehrenfest relation.

Figure 4 Thermal expansion coefficient for the unit cell volume of PuCoGa5 (in an expanded scale around Tc in the inset). Error bars have been estimated as five times the statistical error provided by the Rietveld refinement. If not shown, they are smaller than the data symbol size. Reprinted with permission from Eloirdi et al. (2017). Copyright (2017) by the American Physical Society.

3. Resonant X-ray scattering

Resonant X-ray scattering (RXS) occurs when a photon is absorbed promoting a core electron to empty states, and is subsequently re-emitted when the electron and the core hole recombine (Hannon et al., 1988, 1989). The process introduces anisotropic contributions to the X-ray susceptibility tensor, whose amplitude strongly increases as the photon energy is tuned to an atomic absorption edge. The scattering amplitude also depends on the initial and final polarization of the photons. Measurements are usually performed with incident photons linearly polarized along the direction perpendicular to the scattering plane (σ polarization), whilst a polarization analyser is used to detect photons linearly polarized either along the same direction (σσ channel) or parallel to the scattering plane (σπ channel).

The first RXS measurements were performed with photons tuned to the L edges of holmium metal (Gibbs et al., 1988), and it was soon realized that to observe the maximum magnetic effects one had to tune to energies that were associated with empty states of the spin-polarized electrons, i.e. L edges for 3d systems, and M edges for 4f and 5f systems. At the M₄ edge of uranium, in the compound UAs (Isaacs et al., 1989) an increase in intensity of six orders of magnitude was observed for magnetic Bragg peak when the material ordered magnetically.

For an electric dipole transition (E1), involving the excitation 3d_3/2,5/2 → 5f at the M_4,5 absorption edges of an actinide atom, the resonant X-ray scattering amplitude contains a scalar term, probing the electric charge, a rank-1 tensor odd in time reversal symmetry, probing the magnetic dipole moment, and a rank-2 tensor even under time reversal, probing the electric quadrupole moment,

where ε_i,f are unit vectors giving the polarization of the incident (i) and scattered (f) photons, is the direction of the magnetic dipole moment, is a rank-2 tensor proportional to the electric quadrupole operators or arising from an intrinsic asymmetry of the crystal lattice, and F_1q (q = 0, ±1) are resonant energy factors (Hill & McMorrow, 1996).

If magnetic dipole moments or electric quadrupoles order, and the photon energy is large enough for diffraction to occur, the interference of the anomalous scattering amplitudes leads to the appearance of Bragg peaks at positions Q forbidden by the crystallographic space group. Their intensity depends to a marked extent on the energy E of the incident photon across the M_4,5 actinide absorption edge, on the polarization of the incident and diffracted photons, and on the sample rotational orientation around the scattering vector Q (azimuthal angle Ψ).

In the case of electric quadrupole order, the structure factor is obtained from the electric quadrupole operators in Cartesian components, = (J_i J_j+J_j J_i)/2 (ijk = xyz) and J being the angular momentum operator), as

where the sum runs over all the atoms in the unit cell, at positions r_n, and the scattering amplitude at resonance conditions is = .

RXS experiments provide direct evidence for the ordering of electric quadrupole moments in UO₂ (Wilkins et al., 2006), NpO₂ (Paixão et al., 2002; Caciuffo et al., 2003), and in mixed U_1–xNp_xO₂ solid solutions (Wilkins et al., 2004). These oxides crystallize in the face-centered-cubic (f.c.c.) fluorite structure, but in the ordered phase resonant superlattice Bragg peaks appear at positions that are forbidden in the space group, such as Q = (00ℓ), ℓ = 2n + 1 (Fig. 5). The nature of the order parameter can be established by analyzing the azimuth angle dependence of their intensity in different polarization channels. As shown by equation (1), the term probing magnetic dipoles rotates the photon polarization by π/2. If measurements are performed with σ-polarized incident photons, magnetic scattering appears only in the σπ channel, whereas quadrupole scattering will contribute to both σπ and σσ channels. The intensity modulation of the forbidden peaks provides information on the relative orientation of the moments carried by the atoms in the base of the unit cell.

Figure 5 The integrated intensity of the (003) superlattice Bragg peak as a function of the photon energy around the M5 and M4 absorption edge of U and Np in UO2 and NpO2. Data were collected in the ordered phase, with the incident beam polarized perpendicularly to the diffraction plane (σ) and the scattered photon beam polarized in the diffraction plane (π). The intensity data are corrected for self-absorption. The maxima of the intensity enhancement occur at the E1 dipole threshold energy and are associated with 3d3/2,5/2 → 5f transitions. Note that the shape of the M4 resonance in NpO2 is a Lorentzian squared, as predicted by Nagao & Igarashi (2005). The inset on the upper side is a schematic representation of the resonant scattering process. The central inset is a sketch of the scattering geometry used to measure the integrated intensity as a function of azimuthal angle Ψ, describing the rotation of the sample about the scattering vector. The arrows ε(i,f)(σ, π) correspond to the polarization direction of photons polarized perpendicularly (σ) or parallel (π) to the scattering plane. The u3 Cartesian axis is antiparallel to scattering vector Q.

UO₂ orders at T_N = 30.8 K. The primary order parameter is the magnetic dipole, whilst electric quadrupoles act as secondary order parameters. A 3-k, type-I, transverse antiferromagnetic structure with propagation vector k = (001) becomes stable below T_N (Burlet et al., 1986; Blackburn et al., 2005). The symmetry of the lattice is reduced to Pa3 and the uranium sublattice becomes simple cubic. Each of the four atoms in the base (C_2h point group) carries an electric quadrupole moment given by a linear combination of the three Γ₅ quadrupoles transforming as xy, xz, and yz. The resulting order is also transverse 3-k. The two possible symmetry-equivalent S domains are shown schematically in panels (b) and (c) of Fig. 6. A visual inspection of the figure makes immediately evident that the order of the electric quadrupoles must be accompanied by an internal distortion of the oxygen sublattice as a consequence of the perturbed electrostatic interaction between the oxygen anions and the asymmetric 5f electronic cloud around the uranium ions. This distortion of the oxygen atoms in UO₂ was first reported (Faber et al., 1975) in 1975 using neutron scattering, but the reason for the distortion was not understood at that time. Neutrons cannot observe the electric quadrupoles.

Figure 6 Schematic representation of the projection onto the ab plane of the 3-k magnetic and electric quadrupole ordering for the longitudinal (a) configuration and the two S-domains of the transverse configuration (b, c). The magnetic dipole moments are represented by blue arrows whereas the electric quadrupole moments are shown as green ellipsoids. The red spheres represent oxygen atoms. UO2 adopts the transverse structure, whereas the electric quadrupole longitudinal order is realized in NpO2 with zero magnetic dipole moment. Reprinted with permission from Wilkins et al. (2006). Copyright (2006) by the American Physical Society.

In NpO₂, a second order phase transition is observed at T₀ = 25 K. In this case, the crystallographic structure is preserved and neither external nor internal distortions are observed (Caciuffo et al., 1987). The Ψ dependence of the (001) and (003) Bragg peaks intensity has been measured with the sample kept at T = 10 K at the maximum of the M₄ absorption edge (E = 3.846 keV) (Paixão et al., 2002). Data collected in the σπ and σσ channels have been used to obtain the Stokes parameters

Fig. 7 shows the experimental results. The lines in the figure correspond to calculations assuming the longitudinal 3-k order of Γ₅ electric quadrupoles shown in panel (a) of Fig. 6, assuming a zero ordered magnetic dipole moments (Caciuffo et al., 2003).

Figure 7 Azimuthal angle dependence of the Stokes parameters P1 and P2 for the (001) and (003) superlattice reflections measured in NpO2 at 10 K with E = 3.846 keV. The origin of the azimuthal angle Ψ corresponds to the a-axis lying in the scattering plane. Lines are calculations based on a longitudinal, 3-k order of Γ5 electric quadrupoles with zero ordered magnetic dipole moments. No fitting parameter are involved. Republished with permission of IOP Publishing from Caciuffo et al. (2003); permission conveyed through Copyright Clearance Center, Inc.

For a Q = (0 0 ℓ) reflection, choosing Ψ = 0 where the [100] vector is in the scattering plane with a component parallel to the incident photon beam, the azimuthal and polarization dependence of the RXS amplitude for the considered structure are

where Φ is the quadrupole order parameter. Therefore,

The agreement between experimental and calculated data in Fig. 7 is excellent, considering that no fitting parameters are used. As a consequence of the quadrupole order, the point symmetry at the Np site is reduced to D_3d and the space group is lowered to . However, with Np ions on 4b and O ions on 2a and 6d Wyckoff positions, the crystallographic extinction rules remain the same as those of the space group. Also for NpO₂ quadrupoles are not the primary order parameter. Indeed, by probing the dynamics of the ordered state by inelastic neutron scattering it appears that the driving order parameter is provided by the rank-5 magnetic tri­akontadipoles (Santini et al., 2006; Magnani et al., 2008).

Searches for quadrupolar order by RXS experiments have been performed also on URu₂Si₂. This intermetallic compound has been widely investigated (Mydosh et al., 2020) in an attempt to explain the nature of its phase transition at T₀ = 17.5 K. The puzzle arises from the difficulty in reconciling the tiny value of the ordered magnetic moment (μ₀ ≃ 0.03μ_B along the c-axis of the tetragonal unit cell) with the large macroscopic anomalies observed at T₀ (Broholm et al., 1991; Mason et al., 1990; Walker et al., 1993). For instance, if the order parameter were the magnetic dipole moment, the anomaly in the specific heat should be about 30 times smaller. This indicates that macroscopic anomalies are not associated with μ₀, but rather with a hidden order parameter not directly coupled to scattering probes. Among a number of theoretical models (Oppeneer et al., 2011; Mydosh et al., 2020), a staggered ordering of electric quadrupoles has been suggested to occur in URu₂Si₂ (Santini & Amoretti, 1994; Santini et al., 2000).

RXS experiments at the uranium M₄ absorption edge have not confirmed this hypothesis (Walker et al., 2011). Data have been collected on a high-quality single crystal of URu₂Si₂, cut with a [101] direction specular, spanning an extended region of the reciprocal space plane [H0L]. The results exclude electric quadrupoles of any symmetry as a hidden-order parameter with a propagation vector in the explored region. Indeed, as shown in the top panel of Fig. 8, forbidden Bragg peaks emerging in the ordered state have non-zero intensity only in the σπ polarization channel, with an azimuthal angle dependence corresponding to an ordered magnetic dipole moment along the crystallographic c-axis. The bottom panel of Fig. 8 shows that the Ψ dependence of the superlattice peak intensity is very sensitive to the inclusion of a component of the ordered magnetic moment in the ab plane, whose estimated upper limit is ∼0.003μ_B. This experimental result essentially eliminates the idea of hastatic order proposed by Chandra et al. (2015).

Figure 8 Top panel: azimuthal dependence of the (201) Bragg reflection in URu2Si2. Solid (red) circles: σπ channel; black squares: σσ channel. The full (red) line is the theoretical intensity variation for magnetic dipoles ordered along [001]. The dashed red (dashed blue) line is the σπ (σσ) Ψ dependence of the intensity expected for xy quadrupole order. Bottom panel: experimental data compared with theoretical predictions for a dipole magnetic moment with an increasing component in the ab crystallographic plane. Reprinted with permission from Walker et al. (2011). Copyright (2011) by the American Physical Society.

The capability of the RXS technique to detect higher-order electric multipoles has, of course, been of great interest and further examples are given by Santini et al. (2009). Important early work was done on UPd₃ (McMorrow et al., 2001; Walker et al., 2006), and certainly more materials will be discovered with such quadrupole ordering. An example of Templeton scattering has recently been discovered in a rather complicated actinide system U₂N₃ (Lawrence Bright et al., 2019), and, since it was observed at the M₄ edge of uranium, this suggests a participation of the 5f states in the bonding of this material, although the precise form has not yet been determined.

The enormous enhancement of the magnetic signal at the actinide M edges has also been of interest in many actinide materials to learn more about the magnetic structure itself. Early work, for example, determined in greater detail the nature of the multi-k structures found in materials with transuranium ions (Langridge et al., 1994a,b; Normile et al., 2002a,b; Lidström et al., 2000), as well as those with uranium (Longfield et al., 2002; Bernhoeft et al., 2004a), and thin films (Bernhoeft et al., 1998; Bao et al., 2013). The large intensity also allowed experiments to probe surface magnetism in UO₂ showing new examples of surface ordering not previously measured (Watson et al., 1996, 2000; Langridge et al., 2014). Similarly, the large intensity led to the discovery of an interesting effect that appears to occur when a magnetic material disorders. No satisfactory explanation of this effect has yet been proposed (Bernhoeft et al., 2004b). After a considerable effort this small effect was also seen with neutrons in the antiferromagnet MnF₂, showing that it is not limited to actinides, and is not related to a surface effect (Prokeš et al., 2009).

4. X-ray absorption spectroscopy

X-ray absorption spectroscopy (XAS) of actinides at synchrotron radiation sources started in the late 1980s, exploring both the L_2,3 (2p → 6d) (Kalkowski et al., 1987a; Bertram et al., 1989) and the M_3,4,5, N_4,5, and O_4,5 (nd → 5f; n = 3,4,5) (Kalkowski et al., 1987b) absorption edges. The potential of XAS for elucidating the electronic structure of actinides was immediately recognized and a number of studies on actinide speciation in compounds and minerals followed (Silva & Nitsche, 1995; Conradson et al., 1998; Denecke, 2006). This body of experimental work stimulated the construction of dedicated beamlines, such as CAT-ACT at ANKA (Dardenne et al., 2009; Zimina et al., 2017) and ROBL-II at ESRF (Scheinost et al., 2021). Today, XAS experiments at state-of-the-art spectrometers have been extended to the heaviest actinides available in macroscopic quantities, as recently demonstrated by a study of a coordination complex of ²⁵⁴Es that used less than 200 ng of this highly radioactive isotope (half-life of ∼275.7 days) (Carter et al., 2021).

Besides the scientific importance, actinide speciation studies are key to understanding how these elements can contaminate the environment and how best they can be removed. The clean up of the Rocky Flats Nuclear Weapons Plant, near Denver, USA, is exemplary of the practical importance of these studies (Clark et al., 2006).

From the point of view of physics, XAS experiments at the M_4,5 and N_4,5 absorption edges of actinides have been important to probe the relativistic nature of the 5f electrons, thanks to the application of simple spin–orbit sum rules relating the branching ratio of the core-valence transitions to the expectation value of the angular part of the 5f spin–orbit interaction per hole (Thole & van der Laan, 1988; van der Laan et al., 2004; Shim et al., 2009; Caciuffo et al., 2010a). The information obtained is analog to that provided by electron energy-loss spectroscopy experiments (EELS) (Moore & van der Laan, 2009), a technique that proved to be very useful in elucidating the change between α and δ plutonium (Moore et al., 2006), the magnetic stabilization in curium metal (Moore et al., 2007), or to study rare metals only available in small quantities (Dieste et al., 2019).

5. X-ray magnetic circular dichroism

XMCD is associated with time-reversal symmetry breaking by a magnetic field and involves electric dipole or electric quadrupole transitions promoting an electron in a spin–orbit split core state to an empty valence state of the absorbing atom. The technique provides an element- and shell-specific probe for studying the electronic structure of a wide range of materials (van der Laan, 2013). In actinides materials, XMCD spectra are conveniently measured at the M_4,5 (3d_3/2,5/2 → 5f) absorption edges that directly interrogate the 5f shell. The XMCD signal is obtained as the difference = of the absorption spectra of circularly polarized photons with helicity parallel [μ⁺(E)] and antiparallel [μ⁻(E)] with respect to a magnetic field applied along the incident beam direction. The spectra must be corrected for self-absorption effects and for the incomplete polarization of the incident beam emerging from the crystal monochromator. Standard protocols have been developed for performing such corrections (Goulon et al., 1982; Tröger et al., 1992; Pfalzer et al., 1999).

The power of the technique comes from the simplicity of the sum rules relating orbital and spin moments of the absorbing atoms to linear combinations of the dichroic signal integrated over the pair of spin–orbit-split excitations, , normalized to the isotropic X-ray absorption spectrum (Thole et al., 1992; Carra et al., 1993; van der Laan & Thole, 1996; van der Laan et al., 2004),

where I_M4,5 is the integrated intensity of the isotropic X-ray absorption spectrum, and n_h is the number of holes in the 5f shell. The magnetic dipole term T_z is the component along the quantization axis of an operator associated with charge and magnetic anisotropy (Collins et al., 1995) and correlating spin s_i and position r_i of individual electrons, T = . The orbital and spin components of the total magnetic moment of the 5f-shell, μ = −(〈L_z〉 + 2〈S_z〉), can then be obtained from XMCD spectra, together with an estimate of 〈T_z〉, if the value of the total moment μ and the occupation number of the 5f shell are known.

Furthermore, the expectation value of the angular part of the valence states spin–orbit operator, , can be obtained from the XAS branching ratio, B = I_M5/(I_M5+I_M4), as (Thole & van der Laan, 1988)

where Δ is a quantity dependent on the electronic configuration (van der Laan et al., 2004) (for Np³⁺, for instance, Δ = −0.005).

The XAS and XMCD experiments reviewed here were carried out at the ID12 beamline of the European Synchrotron Radiation Facility (ESRF) using samples encapsulated in an Al holder with Kapton windows of 60 µm thickness in total. The available cryomagnet affords magnetic fields up to 17 T and a base temperature of about 2 K. The minimum mass of the sample depends on the magnetic susceptibility of the system. For elemental curium, data have been collected on a 0.55 mg sample of ²⁴⁸Cm (Lander et al., 2019), whereas a sample mass of 16 µg was sufficient for Am in a sample of AmFe₂ (Magnani et al., 2015).

Fig. 9 shows an overview of the XMCD signals measured at the M₅ and M₄ edges in ferromagnetic AnFe₂ (An = U, Np, Pu, Am) compounds (Wilhelm et al., 2013; Magnani et al., 2015). These spectra are compared with the XMCD signal measured for elemental curium (Lander et al., 2019). For convenience, the photon energy is set to zero at the M₅ edge and the amplitude at the M₄ edge is normalized to unity. It must be noticed that the spectral intensity at M₄ actually becomes smaller with increasing atomic number from uranium to americium, as the number of holes in the j = 5/2 subshell decreases. Moreover, in absolute units the XMCD signal for AmFe₂ is smaller than for NpFe₂, reflecting the difference between the magnetic moments in the two compounds, whereas the narrower linewidth of the M₄ line in AmFe₂ indicates the presence of localized 5f states, as observed in PuSb (Janoschek et al., 2015).

Figure 9 XMCD spectra as a function of photon energy through the An M4,5 edges in AnFe2 (An = U, Np, Pu, Am) and elemental Cm. The spectra have been corrected for self-absorption effects and incomplete circular polarization of the incident beam. Adapted from Lander et al. (2019).

In the case of curium, a visual inspection of the XMCD spectra is sufficient to realize that the integrated intensity of the features at M₅ and M₄ are not equal. From the first sum rule, this means that the orbital moment is not zero. The value provided by the experiment at 70 K is μ_L = −〈L_z〉 = 0.10 (1)μ_B, with a ratio μ_L/μ_S = +0.06 close to the calculated value of +0.052 for the f⁷ configuration in intermediate coupling (van der Laan & Thole, 1996). The fact that μ_L and μ_S are parallel in Cm metal is the reason why the sign of the M₄ line suddenly changes in comparison with the earlier elements (Fig. 9).

Equation (7) shows that the spin component of the magnetic moment can only be obtained if the value of 〈T_z〉 is known. This quantity cannot be measured directly. However, it can be immediately obtained from an analysis of the XMCD spectra if an independent measurement of the total magnetic moment μ = μ_S + μ_L is available, for instance from neutron diffraction experiments or, in the case of Np compounds, from ²³⁷Np Mößbauer spectroscopy. Fig. 10 shows the 3〈T_z〉/〈S_z〉 ratio obtained for several compounds for which such information was available. By changing the 5f occupation number n_f, the ratio 3〈T_z〉/〈S_z〉 varies as predicted by atomic calculations in the intermediate coupling (IC) approximation. The correlation is very convincing and suggests that 〈T_z〉 can be reliably estimated by IC calculations when the total magnetic moment μ is not known.

Figure 10 The ratio 3〈T〉z/〈S〉z between the expectation values of the magnetic dipole and the spin operators along the quantization direction as a function of the 5f occupation number nf. Symbols are experimental data obtained from an analysis of XMCD spectra at the M4,5 actinide edges for compounds with different physical properties and crystallographic structure (circles for C-15 cubic Laves phases, squares for orthorhombic structures, down triangles for NaCl-type structure, and up triangles for hexagonal lattice groups). Theoretical estimates are shown for intermediate coupling (IC) (solid lines), Russell–Saunders (LS) (dashed line), and j–j (dotted line) coupling approximations. Reprinted with permission from Magnani et al. (2015). Copyright (2015) by the American Physical Society.

This is, for instance, the case when the absorbing atom is located inside a vortex in a type-II superconducting compound. As an example, Fig. 11 shows XANES and XMCD spectra measured in PuCoGa₅ (Magnani et al., 2017), a tetragonal heavy fermion superconductor with a critical temperature T_c = 18.5 K (Sarrao et al., 2002). The origin of superconductivity in this compound remains puzzling, despite years of intensive investigations (Jutier et al., 2008; Das et al., 2015; Ramshaw et al., 2015). The Pu ground state is non-magnetic (Hiess et al., 2008), and the superconducting order parameter has d-wave symmetry (Daghero et al., 2012).

Figure 11 XANES (solid black lines) and XMCD spectra as a function of photon energy through the Pu M5 (red line) and M4 (blue line) edges in PuCoGa5. Reprinted with permission from Magnani et al. (2017). Copyright (2017) by the American Physical Society.

XMCD spectra have been measured above and below T_c on a 2 mg single-crystal sample of ²⁴²PuCoGa₅ (99.99 wt% ²⁴²Pu) (Magnani et al., 2017). The c-axis of the crystal was oriented along the incident photon beam, and therefore parallel to the applied magnetic field. The critical field B_c2(T = 0) along the c-axis is 63 T, so that a vortex phase is present in an applied field smaller than 15 T. The 5f shell of the Pu atoms inside the vortex cores is polarized by the external field and the XMCD response is different from zero. Applying the sum rules, the moments and their ratio can be extracted from the experimental data. The values obtained are in excellent agreement with dynamical mean field theories (DMFT) predictions of a non-magnetic ground state for Pu in PuCoGa₅ [see Table I of Magnani et al. (2017)]. This surprising result is a consequence of two effects: intermediate valence driven by 5f–6d Coulomb interaction mixes a magnetic 5f⁵ sextet with a non-magnetic 5f⁶ singlet, reducing the magnetic moment; a complete quenching is then produced by a Kondo-like screening promoted by the hybridization between 5f and conduction electron states (Pezzoli et al., 2011; Shick et al., 2013).

6. Non-resonant inelastic X-ray scattering

Non-resonant inelastic X-ray scattering (NIXS) at the O_4,5 (5d_3/2,5/2 → 5f) absorption edges is a bulk-sensitive technique exploiting multipole transitions from core 5d to valence 5f states. For small values of the scattering vector, Q, the NIXS spectra are dominated by the dipole-allowed transitions encapsulated within the giant Fano resonance. For high Q values, the intensity of the dipole transitions becomes negligible and the spectral response is dominated by dipole-forbidden transitions appearing as well resolved multiplet structure in the pre-edge region (Macrander et al., 1996; Gurtubay et al., 2005; Larson et al., 2007; Haverkort et al., 2007; van Veenendaal & Haverkort, 2008; van der Laan, 2012a). These features provide information on states with symmetries different than those interrogated by electric dipole transitions in XAS, and are sensitive to atomic environment, valence, and hybridization effects (Gordon et al., 2008).

Compared with other X-ray spectroscopy techniques, NIXS is intrinsically intensity-limited. On the other hand, the weakness of the involved photon–matter interaction process and the absence of an intermediate state make it easy to model the experimental data in a quantitative way. Another advantage of NIXS is that shallow energy edges are probed by hard X-rays. Bulk-sensitive information is, therefore, obtained. Moreover, the high penetration depth of hard X-rays (∼5 µm in UO₂ for 10 keV incident photons) makes feasible the study of encapsulated samples, which is a prerequisite in the case of transuranium materials.

Far away from resonance conditions, the radiation–matter interaction is dominated by a term proportional to the square of the vector potential A. Within the first Born approximation, the NIXS double differential cross section is obtained by expanding the transition operator in terms of spherical harmonics, and is given by (Schülke, 2007; Haverkort et al., 2007; Gordon et al., 2008)

where Q = k_i − k_f is the scattering vector, k_i,f are the wavevectors of the incident and scattered radiation, D_k,m is a term describing the angular dependence of the signal, and j_k(Qr) are spherical Bessel functions of rank k. The multipole moments k for the ℓ → ℓ′ shell transition are limited by the triangular condition, |ℓ − ℓ′| ≤ k ≤ ℓ + ℓ′, and the parity rule, ℓ + ℓ′ + k = even. Thus for d → f transitions, only k = 1 (dipole), k = 3 (octupole), and k = 5 (triakontadipole) transitions are allowed. The interrogated final-state J′ and L′ are restricted by the conditions |J − k| ≤ J′ ≤ J + k and |L − k| ≤ L′ + ΔS ≤ L + k. Here, ΔS = 0 in LS coupling and ΔS = ±1 in intermediate coupling. Lower energy states with high L′ and J′ values can therefore be reached by higher multipole transitions.

The potential of NIXS for characterizing the dynamical electron-density response in actinide materials has been demonstrated in a few studies of oxides and intermetallic compounds at the O absorption edges (Caciuffo et al., 2010b; Bradley et al., 2010; Sen Gupta et al., 2011; Sundermann et al., 2016, 2018, 2020). Attempts to carry out NIXS experiments at the N edges of actinides have been frustrated by the weakness of the signal.

Fig. 12 shows, as an example, the NIXS room-temperature spectra obtained for UO₂ in an energy loss (ℏω = E_i − E_f) range encompassing the uranium O_4,5 absorption edges (Caciuffo et al., 2010b). Data were collected at the ID16 inverse-geometry, multiple-analyzer-crystal spectrometer (Verbeni et al., 2009) of ESRF, with a resolution of ∼1.3 eV at a final photon energy E_f of 9.689 keV. The sample was a UO₂ single crystal with the exposed external surface perpendicular to [111].

Figure 12 NIXS spectra measured for UO2 at the uranium O4,5 edges with a fixed final energy of 9.689 keV, at different values of the scattering vector Q. Data at different Q values are normalized to the peak intensity of the feature centered at about 104 eV. Reprinted with permission from Caciuffo et al. (2010b). Copyright (2010) by the American Physical Society.

At Q = 2.81 Å⁻¹, the response is dominated by the dipole transition and shows the broad, asymmetric Fano profile due to resonant decay into continuum states, as observed by XAS measurements (Kalkowski et al., 1987b). Above Q ≃ 9 Å⁻¹, the higher multipole transitions appear at lower energies, with three resolved features at 94.9, 97.3, and 103.6 eV at Q = 9.88 Å⁻¹, providing a fingerprint for the uranium ground state (the radial and angular dependence of the NIXS cross section for the O-edge of a U⁴⁺ 5f² configuration is discussed by Sundermann et al. (2016) and Caciuffo et al. (2010b).

An excellent agreement was found between the experimental data shown in Fig. 12 and many-electron atomic spectral calculations in intermediate coupling that allows one to identify the origin of the observed multipole transitions (Caciuffo et al., 2010b, 2012a). Similar experiments were also extended to transuranium materials, investigating 5f³, 5f⁴, and 5f⁵ configurations in NpO₂, PuO₂, and β-Pu₂O₃ (Sundermann et al., 2020).

Contrary to the dipole transition, the multipole transitions in the NIXS process branch to representations with distinct angular dependence. This fact introduces an anisotropy of the NIXS signal, with measurable differences of the spectral intensity along different directions of the momentum transfer. Such a directional dichroism can be exploited for gaining information about symmetry and strength of the crystal-field acting on the scattering atoms (Willers et al., 2012). The potential of the method has been demonstrated for the tetragonal URu₂Si₂ intermetallic (Sundermann et al., 2016) and for cubic UO₂ (Sundermann et al., 2018). Fig. 13 shows the comparison between experimental data and simulations in the case of UO₂.

Figure 13 Difference between the experimental NIXS spectra measured at the uranium O4,5 edges in single-crystal UO2 with momentum transfer direction = along [001] and [111]. Calculations assuming a Γ5 triplet crystal field ground state and a crystal field potential strength compatible with inelastic neutron scattering experiments are shown by the solid line. Reprinted with permission from Sundermann et al. (2018). Copyright (2018) by the American Physical Society.

The sum rule for the branching ratio of the electric multipole transitions from a core hole to a spin–orbit split valence state probed by NIXS has been derived by van der Laan (2012b). It shows that the effect of the valence spin–orbit interaction on the branching ratio depends on k, the rank of the transition. These effects are very large at the end of the transition series, as in the case of the Er 4f¹¹ M_4,5 spectra, but are observable also for light actinides, as for the U 5f² O_4,5 transition, providing additional information about the electronic structure of the investigated material (van der Laan, 2012b).

7. Resonant inelastic X-ray scattering

Resonant inelastic X-ray scattering (RIXS) experiments consist of measuring the energy dependence of the emission line after creating a core hole by photons tuned around an atomic absorption edge (Kotani & Shin, 2001; Ghiringhelli et al., 2004; Braicovich et al., 2010; Rueff & Shukla, 2010; Ament et al., 2011). As shown schematically in Fig. 14, RIXS is a two-step process involving first the absorption of a photon and the promotion of a core electron into an empty valence state, followed by the emission of a photon with smaller energy and the filling of the core hole by an electron sitting either in a core level [direct core-to-core RIXS, Fig. 14(a)] or in the occupied valence band [direct valence-to-core RIXS, Fig. 14(b)]. The process therefore probes the convolution between empty (absorption step) and occupied (emission step) density of states and leaves an electron–hole excitation in the final state. A different RIXS process is also possible, upon which the electron ejected from the core level and promoted above the Fermi energy refills the core-hole, after the creation of a hole–electron excitation in the valence band [indirect RIXS, Fig. 14(c)].

Figure 14 Schematic representation of direct (a, b) and indirect (c) RIXS processes.

In a RIXS experiment, the scattering cross-section is measured by scanning the incident and emitted photon energy plane in a region around the selected absorption edge and emission transition. In the case of actinides, the combinations used in the majority of the experiments performed so far are the L₃ − Lα₁(Lβ₅) and the M_4,5 − M(α,β) ones. The former, spanning the incident energy range ∼17–19 keV and an energy transfer range ∼3–4 keV, implies an absorption transition 2p⁶6dⁿ → 2p⁵6dⁿ⁺¹ followed by the emission transition 3d¹⁰(5d¹⁰)6dⁿ⁺¹ → 2p⁶3d⁹(5d⁹)6dⁿ⁺¹. The latter, in the incident energy range ∼3.5–4.3 keV and energy transfer range ∼0.3–0.4 keV, probes the 3d¹⁰5fⁿ → 3d⁹5fⁿ⁺¹ absorption followed by a 4f¹⁴5fⁿ⁺¹ → 3d¹⁰4f¹³5fⁿ⁺¹ emission.

Incident energy scans at the maximum of the emission line are called high-energy resolution fluorescence detected X-ray absorption near-edge structure (HERFD-XANES) spectra. On the other hand, measurements at fixed incident energy and varying final energy are referred to as resonant X-ray emission spectroscopy (RXES) (Rueff & Shukla, 2010; Vitova et al., 2010; Kvashnina et al., 2014). Compared with conventional XANES, HERFD-XANES offers the advantage of an increased resolution in the absorption process, since the final transitions are from intermediate states with larger intrinsic core-hole interaction lifetime. At the U L edge, for instance, the energy resolution is improved from ∼8 to ∼4 eV, whilst energy resolutions of 0.4–0.3 eV can be obtained at the U M_4,5 edges (Kvashnina et al., 2014).

At the L₃ edge and the Lα₁ emission line, HERFD-XANES and RXES have been used to study the multiconfigurational nature of 5f orbitals in several uranium and plutonium intermetallics and oxides, demonstrating that the edge position can be related to the density of states at the Fermi level and that the contributions from different 5f electron configurations can be estimated from an analysis of the spectral features (Booth et al., 2012, 2016; Tobin et al., 2015).

As both incident and scattered photon energies are high in a RXES measurement at the L₃ edge of actinides, this technique has been applied to study the evolution of the 5f states upon compression. Being sensitive to final-state energy distributions in the presence of a core-hole, RXES is a good probe of mixed-valency (Rueff & Shukla, 2010). Experiments have been performed on localized magnetic (UPd₃) and heavy fermion (UPd₂Al₃) compounds, showing for the first a stable U⁴⁺ valence state, for the second a progressive transformation upon compression from a mixed valent U^4−δ state to a U^4+δ configuration (Rueff et al., 2007). This kind of measurement in diamond anvil pressure cells requires a very small amount of material, usually (100 µg). This is particularly welcome when working on transuranium materials because it eases the management of the safety risk. For instance, an experiment at the L₃ edge of elemental americium (18.518 keV) was performed using a wide-aperture membrane-type diamond-anvil cell with a ∼5 µm-thick foil of ²⁴³Am metal loaded into the hole (130 µm diameter) of an inconel gasket (Heathman et al., 2010). The energy dependence of the Lα₁ emission line (14.625 keV) was measured for pressure ranging from 4 GPa (double-hexagonal closed-packed Am phase I) to 23 GPa, into the orthorhombic Am IV phase (see Fig. 2). 5f electrons are supposed to be localized in Am-I and itinerant in Am-IV (Heathman et al., 2000; Griveau et al., 2005). Many-body electronic structure calculations, based on the dynamical mean field theory approximation, suggested that the localization delocalization edge is approached by mixing the non-magnetic 5f⁶ Am ground state with the magnetic 5f⁷ configuration, combined with hybridization with the 6d valence band (Savrasov et al., 2006). The experiment, however, does not support such a scenario as no evidence of a mixed valence character emerges at high pressure (Heathman et al., 2010).

RXES measurements at the M_4,5 edges have the advantage over measurements at the L edge of interrogating directly 5f states. The lower energies involved (few keV), however, imply a reduced photon penetration depth (few hundreds of nm) and, therefore, attention must be paid to avoid any surface oxidation of the measured sample. Such experiments have been performed to elucidate the gradual conversion of the U oxidation state in mixed uranium oxides, with the direct observation of U(V) in a binary oxides (Kvashnina et al., 2013; Gouder et al., 2018), to study coordination complexes (Vitova et al., 2010; Bès et al., 2016), and uranium intermetallic compounds (Kvashnina et al., 2017).

Kvashnina et al. (2017) report full core-to-core RXES maps for UPd₃, USb, USn₃, and URu₂Si₂. The experiment was performed at the ID26 beamline of ESRF by scanning the incident energy across the U M_4,5 edges at different emission energies around the M_α and M_β emission lines. For UPd₃, a localized system with 5f² configuration, incident energy scans at the maximum of the emission lines show a shift of the white line by ∼0.2 eV compared with UO₂ (also a localized system with 5f² electron configuration but with an empty conduction band), and that the white line shift for 5f² (UPd₃) and 5f³ (USn₃) in intermetallic compounds is smaller than in ionic compounds. For URu₂Si₂, where an itinerant character of the 5f electrons is expected, the spectral features observed at the M edges suggest almost tetravalent U atoms, with a 5f² configuration. This is in contrast with the analysis of L₃ RXES data that gives a value n_f = 2.87 ± 0.08 for the 5f occupation number (Booth et al., 2016), but in agreement with the conclusions of polarized neutron scattering (Santini et al., 2000; Ressouche et al., 2012), NIXS experiments (Sundermann et al., 2016), and high-resolution RIXS measurements at the U O edges (Wray et al., 2015). The discrepancy with the L₃ RXES conclusions about n_f is probably due to 5f–6d hybridization effects that are more relevant at the L₃ edge, where the 5f shell is interrogated only indirectly.

As written above, experiments at the M edges require clean surfaces, because of the small penetration depth of a few hundreds of nanometers. This requirement is much more stringent at the N_4,5 edges (4d → 5f), where the penetration depth is of a few nanometers only. The reward for the efforts required is an energy resolution that can be as high as 30–35 meV, in an energy range up to 1 eV.

Fig. 15 shows the 15-K RIXS spectrum at the uranium N₄ absorption edge (778 eV), measured at the I21 beamline of the Diamond Light Source on atomically flat, epitaxial UO₂ films (∼100 nm thickness) (Lander et al., 2021). With a resolution of 35 meV, the crystal field excitations within the ³H₄ ground state multiplet are clearly resolved between ∼140 and 180 meV, confirming earlier inelastic neutron scattering (INS) studies (Amoretti et al., 1989). INS measurements on UO₂ failed to detect non-dipolar, higher-energy intermultiplet excitations. RIXS, instead, shows excitations at 520–580 meV due to transitions towards components of the ³F₂ excited multiplet. Measurements with π-polarization at the N₅ edge (not shown in Fig. 15) also establish a peak at 920 meV due to a transition to the ³H₅ multiplet. RIXS experiments around the O edges (5d → 5f) have been attempted to determine the oxidation state of curium in oxide forms (Kvashnina et al., 2007). At this edge, however, it is very complicated to access bulk properties by RIXS measurements.

Figure 15 U N4-edge RIXS spectra of UO2 at T = 15 K. Data were taken with a photon energy of 778 eV for σ linear polarization, as shown in the inset. The red lines show the RIXS spectrum calculated for the U^4+ 5f^2 configuration and CF parameters consistent with inelastic neutron scattering. The blue lines show the underlying multiplet peaks (with no line broadening) of the CF excitations around 180 meV and ^3F2 multiplet around 550 meV. Adapted from Lander et al. (2021).

8. High-resolution inelastic X-ray scattering

High-resolution inelastic X-ray scattering (IXS) at third-generation synchrotron radiation sources is a well established technique for mapping phonon branches with meV energy resolution. High-performance spectrometers using crystals in near-backscattering geometry and efficient focusing optics (Burkel, 2000) allow one to measure phonon dispersion curves in crystals with volumes as small as 10⁻⁴ mm³ and in epitaxial thin films less than 500 nm thick (d'Astuto et al., 2002; Rennie et al., 2018a). Compared with neutrons, this is a crucial advantage for studying radioactive materials or, in general, systems for which large single crystals are not available, or high pressure limits the crystal size. The intrinsic background in IXS experiments is very low, the energy resolution is decoupled from energy transfer, and the momentum transfer is energy-independent. A drawback of the technique is that the scattering cross-section is proportional to the square of the atomic number, making it challenging to observe the contributions of light atoms to the vibrational spectra.

The ID28 beamline at ESRF is an example of a high-performance IXS spectrometer. In the incident energy range of interest for actinide systems (∼17–24 keV) a flat Si perfect crystal monochromator in back-scattering geometry, temperature controlled in the mK region, affords an energy resolution of about 1.5–3 meV when the analyzer is thermally stabilized to 6 × 10⁻⁴ K. Properly oriented single-crystal diamond slabs provide ideal windows if sample encapsulation is mandated by safety reasons.

The measurements of the dispersion curves in f.c.c. δ-plutonium performed on large-grain polycrystalline samples are exemplary for illustrating the potential of IXS in the study of actinide materials (Wong et al., 2003). The results, shown in Fig. 16, provide a qualitative validation of the predictions of dynamical mean field theory calculations (DMFT) (Dai et al., 2003), showing that this theoretical approach is appropriate to describe not only the structure but also (at least qualitatively) the dynamics of strongly correlated electron systems. In particular, the experiment confirms the softening of the T[111] modes and the low-shear elastic modulus C′, reflecting the strong coupling between the lattice structure and the 5f valence instabilities.

Figure 16 Phonon dispersion curves at T = 300 K along high-symmetry directions in δ-Pu stabilized by 0.6 wt% Ga alloying (a = 0.4621 nm). The experimental data are shown as circles [black longitudinal (L) modes; red and blue transverse (T) modes]. The branches T1 and T2 propagating along the [0ξξ] direction are polarized along 〈011〉 and 〈100〉, respectively. The solid curves are the fourth-nearest neighbor Born–von Kármán (B-vK) model fit. The dashed curves are calculated dispersions for pure δ-Pu based on dynamical mean field theory (DMFT) (Dai et al., 2003). Adapted from Wong et al. (2003). Reprinted with permission from AAAS.

IXS was also used to measure the phonon density of states of Ga-doped PuO₂ (Manley et al., 2012) and the phonon dispersion in NpO₂, for which available single crystals are far too small for inelastic neutron scattering (Maldonado et al., 2016).

The low-temperature properties of uranium metal were a mystery since elastic constant measurements in the 1960s found most unusual behaviors. The phonons were measured (Crummett et al., 1979) with neutron inelastic scattering in 1979, showing unusual softening along the [100] direction of the orthorhombic structure of α-uranium. Following these measurements, a charge-density wave (CDW) was found to develop at 43 K (Lander et al., 1994). α-U is the only element to exhibit such a CDW at ambient pressure. However, it was not until 2008 that a theory was presented for the phonons (Bouchet, 2008), and this immediately suggested that the phonon anomaly should be suppressed by pressure – a prediction confirmed by experiments (Raymond et al., 2011) with ID28 at the ESRF using pressures up to 20 GPa. Later, experiments with thin films were successful in placing the [100] axis of α-U under tensile stress (Springell et al., 2014), caused by interaction with the substrate, and the results were a CDW developing at 120 K, much higher temperature than in the bulk. The combination of theory and experiment showed the importance of the electron–phonon effects in the metal, and their momentum dependence. Properties such as the equation of state are affected by these considerations (Dewaele et al., 2013).

Recently, the IXS spectrometer has been pushed to new limits by measuring both diffuse scattering and the phonon dispersion curves from a thin (300 nm) epitaxial film of U-Mo alloys. These materials have been of interest for many years, and might find applications as advanced nuclear fuels, but single crystals are not available and there has been controversy over whether the structures are body-centered cubic (b.c.c.) or something more complicated. Growing epitaxial films turned out to be relatively simple, and the subsequent diffuse scattering patterns (Chaney et al., 2021) showed that the structure is essentially b.c.c. but superposed on that symmetry is correlated disorder, where the local symmetry is lower, as if the uranium atoms prefer to have neighbors reminiscent of the low symmetry found in the element at room temperature, and not the high symmetry demanded of b.c.c. The correlation length of such disorder depends on the composition, but is of the order of 5–10 nm.

The phonon dispersion curves, shown in Fig. 17, are close to those calculated by theory for this composition, except that they show large linewidths. The latter are much broader than expected for an alloy, and are due to the correlated disorder in the material. Thus, we see new effects in these materials that have been studied for many years.

Figure 17 Phonon energy (top panel) and linewidth (bottom panel) dispersions on epitaxial U1–xMox thin film alloys. Transverse (longitudinal) acoustic modes are shown as blue squares (red circles) for the alloy with 23 at% Mo. Theoretical results from a virtual crystal approximation for an alloy with 25 at% Mo are shown as dashed white lines. The full spectral function is plotted as a log0.6 color map to rescale the intensity divergence at gamma. All directions are within the parent BZ. The bottom panel shows raw linewidths, Γ0, as gray squares (TA) and circles (LA). Deconvoluted linewidths Γd are shown by dashed blue (TA) and red (LA) trend lines. Reprinted with permission from Chaney et al. (2021). Copyright (2021) by the American Physical Society.

In a similar vein, new experiments (Paolasini et al., 2021) at ID28 on the phonon linewidths of UO₂ below room temperature (this time using a single crystal rather than a thin epitaxial film) have shown that the linewidth broadening in UO₂ is only along the [100] direction, and not in the other directions. Such anisotropic broadening has consequences for the thermal conductivity, which for a cubic material should not be anisotropic.

9. Conclusions

When the first synchrotrons began operations in the late 1970s and 1980s actinides were soon examined, notably with absorption (Kalkowski et al., 1987a; Bertram et al., 1989) and EXAFS measurements (Silva & Nitsche, 1995; Conradson et al., 1998) at the actinide L edges, high-pressure experiments using energy dispersive detectors (Benedict & Holzapfel, 1993), and high resolution XRD to investigate fine details of the diffraction patterns (Grübel et al., 1991). The success of these machines across all parts of the periodic table led, of course, to more powerful synchrotrons being built, and the advantages for the study of actinides increased. They now represent some of the most powerful tools for such research.

This article has focused on experiments performed in the field of condensed-matter physics in the last 20 years at the ESRF in Grenoble, France, with some references to work at other synchrotrons, notably Diamond in the UK, and APS at Argonne National Laboratory (USA). Of course, many experiments have been performed in the domain of chemistry; efforts to study the dissolution of UO₂ in water (Springell et al., 2015; Rennie et al., 2018b) span these two fields. They demand synchrotron radiation, as diffraction signals are required from epitaxial films of <10 nm thickness. We are aware that studies of the physics and chemistry of actinides at the other two third-generation machines, ALS (USA) and SPring-8 (Japan), have been performed and new ones are also underway, and hope that some are covered in similar reviews in this journal.

In concluding this article, we would like to highlight four examples that we believe to have been `game changers' for actinide research.

The first (Section 2) is the capability to observe samples at the µg level, allowing pressures up to 100 GPa to be applied in diamond-anvil cells, and the development of the angular-dependent data collection so sophisticated data analysis can be used to determine the crystallographic structures, see Fig. 2. These experiments, extended to transuranium samples (Lindbaum et al., 2001; Heathman et al., 2005), illustrated the failure of the density-functional theory (DFT) for actinides, and spurred the subsequent work with the dynamical mean-field theory (DMFT) that is frequently mentioned in the text. These experiments could only have been performed on third-generation synchrotrons.

The second example (Section 3) is the discovery in the actinide oxides and UPd₃ of ordering of the electric quadrupolar moments (Santini et al., 2009) below room temperature. In this case intensity is not always the main problem, but complex instrumentation is needed to measure the polarization dependence of the scattered photons, and the azimuthal dependence of the intensity. In the case of NpO₂, for example, the nature of the phase transition at 25 K (Paixão et al., 2002) had been the source of speculation for more than 50 years. How common this phenomenon is in the actinides is still an open question, but the tools to find such effects are now available.

The third example is that the development of the technique of XMCD (Section 4) led to a direct method of determining the absolute values of the spin and orbital moments in actinide materials. Prior to this, the only technique available was using polarized neutrons, but this requires sizable single crystals. Now these quantities can be measured with micrograms of polycrystalline samples. The values, and especially their ratio, are of major importance for actinide research, as with itinerant 5f electrons there is a tendency for a partial quenching of the orbital moment (Lander et al., 1991). Now that the 〈T_z〉 quantities calculated with intermediate coupling (see Fig. 10) can be used, the orbital to spin ratio is readily determined. Fig. 10 also represents perhaps the most convincing evidence given so far that intermediate coupling is a crucial requirement of the physics and chemistry of actinide materials in general.

The fourth example concerns the inelastic scattering of X-rays (Section 7). Here again, the capability to determine the phonon spectra from micrograms of material as at the ID28 beamline has been of great importance. The phonons of δ-Pu [Fig. 16 and Wong et al. (2003)] was a major achievement, especially since the theoretical work was actually published in advance (Dai et al., 2003), and gave the first major `credibility test' to the DMFT theory at that time (2003). More recently, the development of grazing-incidence IXS has led to important new information on the radiation damage in UO₂ (Rennie et al., 2018a) and on the complexity of the U–Mo alloy system (Chaney et al., 2021). The sensitivity of these experiments is outstanding; especially considering the sample mass is less than 100 µg.

The future of actinide research at synchrotrons is clearly a bright one. New machines, and associated complex instrumentation, are coming on-line that will clearly benefit new initiatives. Free-electron lasers (FELs) with pulse durations from a few to hundreds of femtoseconds are now starting to operate, pushing to the femtosecond range the timescale limits of spectroscopic and structure studies (Liermann et al., 2021). Diffraction-limited storage rings (DLSRs) such as MAX-IV (Tavares et al., 2018) and ESRF-EBS (Raimondi, 2016; Chenevier & Joly, 2020), with emittance approaching the diffraction limit and delivering ultra-small X-ray beams, start demonstrating their potential as an extraordinarily powerful tool for the investigation of complex systems and emerging phenomena. These machines will provide an increase in brightness and coherent flux of about two orders of magnitudes, compared with third-generation X-ray synchrotrons, with obvious applications for high pressure studies or coherent diffraction. All such instrumentation will greatly benefit our understanding of this complex row of elements in the periodic table.