# What can studies of the pair-distribution functions (PDF) contribute to our knowledge about UO2?

Lionel Desgranges, Henry Fischer, Gerard Lander, Gianguido Baldinozzi

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Lionel Desgranges, Henry Fischer, Gerard Lander, Gianguido Baldinozzi. What can studies of the pair-distribution functions (PDF) contribute to our knowledge about UO2?. MRS Communications, 2025, 15, pp.441-446. 10.1557/s43579-025-00730-z . hal-05099206

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# What can studies of the pair-distribution functions (PDF) contribute to our knowledge about UO₂?

Lionel Desgranges¹, CEA.DES.IRESNE.DEC, Cadarache, F 13108 St Paul Lez Durance, France

Henry E. Fischer, Institut Laue Langevin, 71 Avenue Des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France

Gerard H. Lander, Institut Laue Langevin, 71 Avenue Des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France; European Commission, Joint Research Centre (JRC), Postfach 2340, 76125 Karlsruhe, Germany

Gianguido Baldinozzi, CentraleSupélec, CNRS, SPMS, Université Paris-Saclay, 91190 Gif-Sur-Yvette, France

Address all correspondence to Lionel Desgranges at lionel.desgranges@cea.fr

(Received 28 February 2025; accepted 29 April 2025; published online: 19 May 2025)

## Abstract

Uranium dioxide has been studied for 70 years but still has some poorly understood properties. We present our last results using neutron total-scattering methods that evidence local deviations from average structure in UO₂ Pair-Distribution Function (PDF). At high temperature, this deviation appears as a decreasing of the UO distance with temperature, which we interpret as a phonon-induced dynamic distortion of uranium's local environment. At low temperature, this deviation takes place in UO₂ magnetic order above the magnetic transition, which is still under study.

## Introduction

Given the fact that UO₂ is the most used nuclear fuel, it is not surprising that it has been investigated over the last 70 years by many different techniques. In this article, we shall give a brief overview of experimental results using neutrons for elastic scattering (Bragg peak refinement) and total-scattering (PDF analysis). Recent reviews of this material include,[¹] which focusses on the properties important for use as a nuclear fuel, and[²] that covers properties measured by neutrons, especially inelastic scattering. Whereas the X-ray scattering length of oxygen is much smaller than that for uranium, such that uranium atoms dominate the X-ray scattering signal, for neutrons the two scattering lengths are comparable. In addition, neutrons are sensitive to the magnetic moments of the uranium 5f states, such that the antiferromagnetic structure of UO₂ was determined as early as 1965[³,⁴] with neutron diffraction. Finally, the energy–momentum dispersion relation for neutrons permits to probe easily the spectrum of vibrational and magnetic excitations. In fact, the phonon dispersion curves of UO₂ were also measured in the 1960 s[⁵] and examined more recently to understand the thermal conductivity in greater detail.[⁶,⁷] An important aspect of UO₂ is that it is strongly anharmonic, a discovery that was first made with diffraction data by Willis and colleagues[⁸] and has been confirmed with neutron inelastic scattering.[⁷,⁹]

Importantly, over the past couple decades has emerged the realization that the average structure of a crystalline solid, i.e., that measured by Rietveld refinement of integrated Bragg peak diffraction intensities, does not always suffice in predicting some physical properties of the sample, such as transport properties and others. Determining the local deviations from this average structure, e.g., local lattice distortions in mixed or

non-stoichiometric crystals, or correlated atomic vibrations, can be important. Such spatial or temporal deviations from the local structure lead to generally ignored diffuse scattering intensity that can, however, be Fourier transformed into a Pair-Distribution Function (PDF). Such a function reveals directly the correlations within these structural deviations. The so-called Total-Scattering measurements can be carried out with either X-rays or neutrons. The accompanying data interpretation using Fourier techniques is known as PDF analysis, which began with liquids/glasses diffraction[¹⁰] in the 1960's, then extended to crystalline solids[¹¹,¹²] and finally to magnetic spin systems.[¹³,¹⁴] Of note is that the real-space function PDF(r) is a model-independent histogram of quasi-instantaneous local interatomic distances in the sample, and the magnetic PDF(r) is a model-independent function showing quasi-instantaneous local spin correlations, both functions being derived from a direct Fourier transform of the total diffraction intensity, including all inelastic and diffuse scattering.

## Structural studies

One of the surprising aspects of UO₂ came from studies of the X-ray PDF(r) at high temperature near the melting point T_m [¹⁵] and confirmed by EXAFS studies at lower temperature.[¹⁶] These studies purported to show that the first nearest-neighbor U–O distance in the structure actually appeared to get shorter as the temperature increased, with somewhat pathological behavior near T_m.[¹⁵] Neither of the two references gave a possible explanation.

Given that the PDF technique observes the local structure of a system, which can deviate in time and space from the average structure probed by Bragg peak refinement, this seemed a good
opportunity to see whether our neutron diffraction measure-
ments could confirm the tendencies reported in Refs. 15 and 16.

As explained in Ref. 17, the neutron PDF(r) as a function
of temperature (Fig. 1) shows quite convincingly that the U–O
bond becomes shorter as the temperature is increased, whereas
the U–U distance increases as expected for a lattice parameter
[see Fig. 2(left) showing the Fm-3m fluorite structure of UO2].
A summary plot in Fig. 2(right) shows that our PDF(r) results
are in good agreement with previous work.[15,16]

The reason for this unusual temperature dependence of the
U–O distance can be traced back to the phonon eigenvectors in
the CaF₂ structure that is shared with UO₂, and this is further
explained in detail in Ref. 17. To summarize briefly, there is
a large portion of the phonon spectra, specifically in the [001]

zone, labeled as the (TO1) Δ₅ mode, where the phonon disper-
sion is small and the energy is close to 30 meV.[⁵,⁶] Because of
the different masses in the structure, the Δ₅ optic phonons have
very little contribution from the uranium atoms and are almost
pure modes involving the oxygen atoms. The Δ₅ eigenvectors
separate the eight equivalent U–O distances into six shorter
and two longer distances, where the longer pair contributes to
a high-r tail observed in the U–O distance distribution (Fig. 1),
and become increasingly anisotropic at higher T. The ensemble-
averaged length of the U–O bond therefore starts to contract as
the Δ₅ mode becomes populated, which occurs gradually up to
about 400 K. For example, the Bose population factor is unity
at 500 K. The effect gets greater as T increases, and, as Ref.
15 shows, becomes pathological near the melting temperature.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Neutron PDF(r) results from 50 to 1023 K. Note the increasing broadening of the U–O peak with increasing temperature as it becomes more asymmetric and acquires a high–r tail, all which suggests that the potential well softens and that anharmonic contributions are present. The r-space resolution is about 0.16 Å (FWHM) for a Qmax = 23.5 Å⁻¹.</figcaption></figure>

**Magnetic correlations**

Although our magnetic PDF studies are not particularly concerned with the antiferromagnetic (AFM) structure *below* $T_N$=30.8 K, we emphasize that much work has already been done on this aspect of UO$_2$. As early as the 1960's, the propagation vector of the ordered AFM structure of UO$_2$ was determined to be $k = <001>$, which implies alternating magnetic moment components on alternating layers of the simple CaF$_2$ fcc structure. A pioneering early study[19] of the spin-wave dispersion relations at 9 K was determined by Cowley & Dolling, and this led to many further studies—most of which are summarized in Ref. 2. In the present work, we are interested in the magnetic spin correlations *above* $T_N$. Upon cooling UO$_2$ to $T_N$, three important events occur: (i) the magnetic moments of 1.74 µ$_B$/U-atom order into the AFM structure and the transition is of first order (as shown in the graphical abstract), (ii) the electric quadrupole moments associated with the U 5f electrons

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Left: The 5.45 Å fcc unit cell (space group Fm-3m) of UO₂ where the O atoms (blue) form a simple cubic sublattice whose cells are alternately empty or contain a U-atom. Right: Experimental results for U–O and U–U distances in UO₂ as determined from various techniques, as a function of temperature. Red symbols refer to U–U distance, left-hand scale. Filled diamonds: data from neutron PDF analysis. Dashed line: values derived from X-ray lattice parameters recommended by Martin.[18] Black symbols refer to U–O distances, right-hand scale. Filled squares: values derived from neutron PDF analysis.[17] Open circles: values derived by X-ray PDF analysis and given by Skinner et al.[15] Open squares: values derived from EXAFS experiments by Prieur et al.[16]</figcaption></figure>
also order,[20] and (iii) the oxygen atoms exhibit an internal
shift,[21] which reduces the symmetry of the space group from
Fm-3m to Pa-3m, but the overall unit cell remains cubic. The
resulting AFM structure has been shown to be triple-k, which
implies that the *k* = <001> arrangement is consistent with three
independent branches of the AFM structure *existing within any*
*given volume* of the chemical structure.[22]

Experiments *above* T<sub>N</sub> have shown that the magnetic diffuse
scattering, representing *dynamic* spin correlations and short-
range order between the disordered magnetic moments, can
be observed up to ~200 K,<sup>[23]</sup> almost seven times T<sub>N</sub>, which
is unusual but not completely unexpected for an fcc geometri-
cally frustrated magnet. A question is whether measuring this
diffuse scattering as a function of temperature can derive the
correlations between different spin pairs, and then perhaps even
determine details about the exchange interactions in UO<sub>2</sub>. A
further question is whether the correlations in the paramagnetic
state are of the 3 k type, or of the 1 k type as suggested in Ref.
23. Note however that in the case of 1 k dynamic local spin
correlations in the paramagnetic state, and even for a single-
crystal sample, the ensemble averaging over magnetic correla-
tion domains would make the total-scattering neutron diffrac-
tion intensity, and hence the magnetic PDF(r), indistinguishable
from that in the case of 3 k dynamic local spin correlations.
Such would also be the case for static spin correlations: the
domain averaging prevents distinguishing 1 k from 3 k, which
would require inelastic neutron scattering measurements like
in Ref. 24.

Our neutron total-scattering measurements[25] were carried
out on the D4 diffractometer[26] at the Institut Laue-Langevin
(Grenoble), which provided not only the atomic PDF(r) data
of Figs. 1 and 2 above, but also permitted *via* temperature sub-
traction to isolate the total magnetic scattering intensity by
removing the nuclear Bragg peaks as well as the paramagnetic

scattering intensity representing only non-correlated spins, as
shown in Fig. 3(left).

The Reverse Monte Carlo (RMC) program Spinvert[27] was
then used to construct a real 3D configuration of magnetic spins
that corresponded to the observed Q-space magnetic scattering
data, and whose Fourier transformed fits were therefore free of
non-physical features, as shown by the final magnetic PDF(r)
curves in Fig. 3(right). Clearly visible are the AFM (negative
peaks) and FM (positive peaks) spin correlations as a function
of inter-spin distance. Remarkably, there are still nearest-neighbor
AFM correlations at 3.85 Å for the highest temperatures,
attesting to the fcc geometrical frustration for AFM interactions
in UO₂. The PDF(r) at low r (below ~3 Å) follows a smooth
upward slope as expected for an AFM structure (see Ref. 14),
although some small Fourier termination ripples can be seen.
Also interesting is the clear indication that the correlations of
spin pairs separated by 6 Å <r<7 Å, as well as by 9 Å <r<11
Å, are qualitatively different for T>Tₙ as compared to T<Tₙ.

It is important to note that the Spinvert program does not
take into account any spin interaction energies whatsoever, but
simply rotates spins one by one until a configuration is found
that reproduces well the observed magnetic diffuse scattering,
rejecting “bad” moves according to a Boltzmann factor cal-
culated from the Chi-square discrepancy with respect to the
data, and as for all RMC algorithms, this leads to maximizing
the entropy of the final spin configuration. We used a cubic
simulation box with periodic boundary conditions and dimen-
sions of 90 Å on a side, corresponding to the neutron coherence
volume[15] for the D4 instrument, which thus guaranteed the
proper Q-space resolution for the simulated data.

Complementary R-space information is provided by the Spin
Correlation Function SCF(r) shown in Fig. 4(left), which is sim-
ply the classical dot-product of two neighboring spin vectors of
unit length as averaged over all spins at a given distance within
the Spinvert simulation box, and thus takes on values between

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Left: Total magnetic scattering from UO<sub>2</sub> as a function of temperature from 10 to 250 K. The nuclear scattering (coming from the atomic structure) has been removed at each temperature by subtracting the total-scattering diffraction pattern obtained at 300 K that represents a paramagnet. Right: Magnetic PDF(r) obtained by direct Fourier transform of the Reverse Monte Carlo fits to the data at left using the program Spinvert.[27]</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Left: Spin Correlation Function SCF(r) for UO₂ as a function of temperature as extracted from the Spinvert 3D spin configurations. Of note is the abrupt decrease in the amplitude of the spin correlations as T>T<sub>N</sub>, and also that the short-range spin correlations persist to higher T than do the long-range correlations. The overall Gaussian-like falloff of the SCF(r) is due to the limited Q-space resolution of the D4 diffractometer and represents the spatial extent of the neutron coherence volume.[15] Right: The temperature dependence of the magnetic spin correlations in UO₂ plotted for each spin pair as a separate curve for T>T<sub>N</sub>. Note that these short-range *dynamic* spin correlations exist well into the paramagnetic state, which is typical for geometrically frustrated magnetic systems such as the fcc lattice of spins in UO₂.</figcaption></figure>

1 (parallel spins) and – 1 (anti-parallel spins). This figure makes clear that the loss of spin correlations as $T > T_N$ is more abrupt for long inter-spin distances than for short inter-spin distances. As compared to the mPDF(r), the SCF(r) as extracted from a simulation box has the advantage of providing exact inter-spin distances and unambiguous identification of the spin pairs, but the mPDF(r) has the advantage of being sensitive additionally to the direction of the inter-spin vector relative to the two spin orientations (see Ref. 14), which of course is not retained by taking a simple dot-product of two spin vectors in space.

By regrouping the SCF(r) results of Fig. 4(left), one can plot the correlation of each spin pair as a function of temperature, as shown in Fig. 4(right). Of particular interest are the spin correlations for $T > T_N$, since these correlations are *dynamic* rather than static, and therefore, they cannot give rise to elastic Bragg peak intensities that could be analyzed by Rietveld refinement. Figure 4(right) shows clearly that the AFM nearest-neighbor (NN) spin correlations (magneta curve) are by far the strongest in the paramagnetic phase and persist until almost room temperature, consistent with the observations of Caciuffo et al.[24] Remarkably, a first-principles dynamical mean-field model of dipole and quadrupolar interactions in UO₂ by Pourovskii and Khmelevskyi [Ref. 28, see their Fig. 4] predicted quantitatively within 20% the same NN spin correlations between U dipoles as extracted by the Spinvert RMC modeling of our data (again, the magenta curve at right in Fig. 4).

## Conclusions and future work

The present brief paper gives a status report on our efforts to understand more about UO₂ using the techniques of atomic and magnetic PDF analysis. We should emphasize that all

the results reported here have been taken on samples subject to powder averaging and are therefore a function of only the magnitudes of the Q- and r-vectors. In addition, our total-scattering diffraction measurements naturally do not distinguish between elastic and inelastic neutron scattering and therefore cannot distinguish, *a priori*, static from dynamic correlations except *via* temperature change of the sample.

Despite these reservations, we have shown that such neutron total-scattering studies can obtain important structural information. Our results confirm the previous work using X-ray PDF[16] and EXAFS studies[17] on UO₂, which indicated a shortening of the U–O bond as a function of rising temperature. Further we postulate[18] that the effect arises from a phonon mode at around 30 meV that separates the U–O bonds into six shorter bonds, and two longer ones; thus, the average value reduces as the temperature is raised and the phonon modes are further populated. We have similar results for CeO₂ and plan to measure also ThO₂. The chances of this happening in other structures, especially those containing heavy and light atoms, would appear highly likely.

Our magnetic PDF(r) and SCF(r) results have so far verified and extended conclusions from previous studies. For example, the abrupt change of the long-range spin correlations at $T_N$ shown in Fig. 4 (left), and in the graphical abstract, has confirmed the first order like transition in UO₂ that was first observed in specific-heat measurements in 1952,[29] as well as the persistence of nearest-neighbor AFM spin correlations to high temperature, consistent with the observations of Caciuffo et al.,[24] which has been quantified in Fig. 4 (right). Most interestingly, our mPDF(r) in Fig. 3(right) shows clear indications that some spin correlations are qualitatively different above $T_N$ as compared to
below $T_N$. We emphasize that magnetic PDF analysis is one of the only quantitative techniques that can probe local spin correlations above the magnetic transition temperature.

Thus, we can answer the question posed by the title of this paper, namely that PDF studies can indeed provide new and complementary information on the atomic and magnetic correlations in $UO_2$, especially as concerns short-range dynamic correlations for $T > T_N$.

We are currently pursuing the interpretation and analysis of our magnetic diffuse scattering data on $UO_2$ along two axes: (1) developing a model of Ornstein-Zernicke-like magnetic “drop-lets,” corresponding to volumes of correlated spins above $T_N$, as a means of describing the observed temperature dependence of spin correlations shown in Fig. 4 (right), and (2) applying the Spinteract program[30] to extract quantitative information from our $T > T_N$ data about the known anisotropy of exchange interactions in $UO_2$, in particular, the coefficients of bilinear terms to be included in the spin interaction Hamiltonian. Preliminary results are very encouraging.

## Acknowledgments

We wish to thank Joe Paddison for his help with the Spinvert simulations and for a critical reading of this manuscript.

## Author contributions

All authors contributed to the study conception and design, and also to data collection. Data analysis was performed by [Henry E Fischer] and [Lionel Desgranges]. The first draft of the manuscript was written by [Lionel Desgranges] and all authors made additional contributions to successive versions of the manuscript. All authors read and approved the final manuscript.

## Funding

Open access funding provided by Commissariat à l'Énergie Atomique et aux Énergies Alternatives. Funding information is not applicable.

## Data availability

My manuscript has associated data in a data repository.

## Declarations

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

### Open Access

This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as

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