Synchrotron x-ray scattering of UN and U2N3 epitaxial films

TL;DR

This study uses synchrotron x-ray scattering to investigate the magnetic ordering and lattice interactions in epitaxial films of UN and U2N3, revealing new insights into their magnetic structures and electron charge contributions.

Contribution

The paper provides the first determination of antiferromagnetic ordering in U2N3 and identifies quadrupolar charge contributions involving 5f electrons, challenging previous assumptions about UN's magnetic distortion.

Findings

Confirmed AF ordering of UN without expected lattice distortion.

Determined AF wave-vector of cubic U2N3 for the first time.

Identified quadrupolar charge contributions in U2N3 involving 5f electrons.

Abstract

We examine the magnetic ordering of UN and of a closely related nitride, U2N3, by preparing thin epitaxial films and using synchrotron x-ray techniques. The magnetic configuration and subsequent coupling to the lattice are key features of the electronic structure. The well-known antiferromagnetic (AF) ordering of UN is confirmed, but the expected accompanying distortion at Tn is not observed. Instead, we propose that the strong magneto-elastic interaction at low temperature involves changes in the strain of the material. These strains vary as a function of the sample form. As a consequence, the accepted AF configuration of UN may be incorrect. In the case of cubic a-U2N3, no single crystals have been previously prepared, and we have determined the AF ordering wave-vector. The AF Tn is close to that previously reported. In addition, resonant diffraction methods have identified an aspherical quadrupolar charge contribution in U2N3 involving the 5f electrons; the first time this has been observed in an actinide compound.

Tables

Table 1. Table 1: Table of structure factors for the first 11 charge reflections in the cubic U 2 N 3 structure; space group # 206. The cubic lattice parameter used is a 𝑎 a = 10.69 Å. The quantities SF U 1 and U 2 are the trigonometric structure factors for the uranium atoms for each reflection. No nitrogen atoms are included (there are 48 in the unit cell), and no scattering factor for the U atoms. The reflections in the first column with an “x” indicate that the reflection is forbidden − - group B; where the group A is with all the atoms in phase, and thus a large intensity. The groups are discussed in the text. In the Observation (Obs.) column, Ma and Ch signify magnetic and charge energy profiles, see Fig. 8 lower panel. The final column gives the origin of the ATS structure factor, see text. Reflections indices here follow the convention for cubic systems, h ℎ h > k 𝑘 k > l 𝑙 l , whereas figures correspond to the observed reflections with the specular direction defined as \hkl [001].

\hkl(hkl)	$\left|Q\right|$ (Å-1)	SF U1	SF U2	Total SF	Group	Observation	ATS
\hkl(000)	0	8	24	32	A
\hkl(110)x	0.831	0.00	0.00	0.00	B		U1	U2
\hkl(200)	1.176	-8.00	7.75	-0.25	C	Ma		U2
\hkl(211)	1.440	0.00	1.99	1.99	D	Ch	U1	U2
\hkl(220)	1.662	8.00	-8.00	0.00	E	Ma		U2
\hkl(310)x	1.859	0.00	0.00	0.00	B	Ma	U1	U2
\hkl(222)	2.036	-8.00	-23.25	-31.25	A	Ch
\hkl(321)	2.199	0.00	-1.99	-1.99	D		U1	U2
\hkl(400)	2.351	8.00	23.01	31.01	A	Ch
\hkl(330)x	2.494	0.00	0.00	0.00	B		U1	U2
\hkl(411)	2.494	0.00	3.85	3.85	D	Ch	U1	U2
\hkl(420)	2.494	-8.00	6.76	-1.24	C			U2

Equations

$$\lambda =2dsin\theta (1-\delta /si{n}^2\theta ),$$

$$\Delta c/c=\delta /si{n}^2{\theta }^{\prime }.$$

$$f=f_0+f^{\prime }+i{f}^{\prime \prime }$$

Full text

††preprint: APS/123-QED

I Introduction

There is renewed interest in uranium nitride as a so-called advanced-technology fuel to replace the current standard fission fuel, UO2. This is principally due to its higher thermal conductivity, 20 W/(m-K) at 1000 K Kurosaki et al. (2000), compared to $\sim$ 3.5 W/(m$\cdot$K) for UO2 Ronchi et al. (2004). In stark contrast to UO2, whose thermal conductivity is entirely driven by the phononic behavior, for UN only $\sim$ 15 % is due to any phonon contribution, and the remainder is electronic Kurosaki et al. (2000). The ability to calculate these electronic contributions and therefore make predictions about the thermal properties is complex, and attempts have been made by Yin et al. Yin et al. (2011) and by Szpunar and Szpunar Szpunar and Szpunar (2014), both of which use approximations. In fact, the electronic structure of UN has been controversial for at least 50 years. Despite many studies, even the number of 5$f$ electron states, and whether they are localized or itinerant (or some mixture), is still being discussed. The work reported here is thus a contribution to these discussions.

We have recently succeeded in preparing thin epitaxial films Lawrence Bright et al. (2018) of UN and a closely associated material cubic U2N3, which is almost always found in conjunction with UN, as it represents a byproduct in the oxidation process. We have also reported the corrosion rates (with H2O2) Lawrence Bright et al. (2019) of these materials, and found that whereas UN is less corrosive than UO2, the U2N3 material is at least 20 times more corrosive than UN. Since U2N3 is found at the surface of UN, this higher corrosion rate is a concern, and our work reported here suggests a possible reason for this difference.

UN has the NaCl fcc cubic structure with $a$ = 4.89 Å at 300 K. The susceptibility gives an effective moment ($\mu_{eff}$) of $\sim$ 2.8 $\mu_{B}$ and a large $\theta_{p}$ of $\sim$ $-$ 300 K in fitting to the Curie-Weiss law. UN orders antiferromagnetically (AF) at TN $\sim$ 53 K with a type-I AF structure with an ordered moment ($\mu_{ord}$) of 0.75 $\mu_{B}$ Curry (1965). The large discrepancy between $\mu_{eff}$ and $\mu_{ord}$ and between TN and $\theta_{p}$ are not understood. Troć et al. (2016) Troć et al. (2016) have recently given an excellent summary of the properties of UN.

Much less work has been done on U2N3, although the structure of the cubic ($\alpha$ form) has been known for many years Rundle et al. (1948), and is the cubic bixbyite structure common to R2O3, where R is a metal atom. The lattice parameter is between 10.6 and 10.7 Å, depending on the exact U/N ratio. Troć (1975) Troc (1975) examined the magnetic properties, and the effective moments are around 2 $\mu_{B}$. The AF ordering temperatures vary as a function of the U/N ratio between 94 K for UN1.50 to $\sim$ 20 K for UN1.72. Neither the type of AF magnetic ordering, nor the ordered moments, are known.

The preparation of such films opens the way for further experimental studies of the properties of both compounds, and thus perhaps a better understanding of the electronic structure, which can then be used in modeling for the thermal conductivity and other properties. In this paper we discuss experiments below room temperature on both UN and U2N3 epitaxial films focused on the interaction of the lattice and the magnetic (electronic) structure. This range of temperature is, of course, irrelevant for nuclear fuel applications, but our emphasis is on the electronic structure and how best to describe it.

II Sample Preparation and Experimental Procedure

Although thin films of UN have been produced before Black et al. (2001); Rafaja et al. (2005); Zhang et al. (2010); Long et al. (2016); Wang et al. (2016); Lu et al. (2016), they have not been epitaxial, but in the best case have been strongly textured Rafaja et al. (2005). As reported in Ref. Lawrence Bright et al. (2018), we used a sapphire \hkl(1-102) substrate with a \hkl[001]-oriented Nb buffer, and the UN grows on this with a 1:$\sqrt{2}$ relation and a 45 ∘ rotation. The growth temperature of the film was 600 ∘C, and a 5 nm cap of Nb was deposited at room temperature on the film to prevent oxidation. The film used at the synchrotron had a thickness of 70 nm and a rocking curve of 1.9 ∘. Thin films of U2N3 have not been reported previously, and we found these can be grown on CaF2 substrates ($a_{0}$ = 5.451 Å) and have a very good mosaic (less than 0.10 ∘) Lawrence Bright et al. (2018) when they are thin. However, for thicker films the mosaic increases, and the 200 nm film we used had a rocking curve of $\sim$ 1 ∘. The lattice parameter in the direction of growth is 10.80(1) Å, compared to $2a_{0}$ (CaF2) = 10.9 Å, and the in-plane parameters were 10.60(2) Å. Based on the atomic volume, this corresponds to an effective cubic lattice parameter of 10.67 Å, suggesting we are close to stoichiometry Troc (1975).

Resonant x-ray scattering (RXS) measurements were conducted at the Materials and Magnetism Beamline I16 at Diamond Light Source Collins et al. (2010). The x-ray energy was tuned to 15 keV ($\lambda$ = 0.8265 Å) for sample alignment and determination of the lattice parameter, due to the increased number of reflections available, and to the uranium $M_{4}$ edge at 3.726 keV ($\lambda$ = 3.327 Å) for measurements of the magnetic diffraction, taking advantage of the resonant enhancement of the magnetic signal. Samples were mounted in a closed-cycle cryocooler for low temperature measurements. A kappa-geometry 6-circle diffractometer provides large access to reciprocal space and the capability of azimuthal scans and grazing incidence diffraction. All measurements were performed in vertical geometry, perpendicular to the incident polarization of the beam and the azimuthal zero reference is taken when the crystal \hkl(001) direction intersects the scattering plane. Scattered x-rays were measured using either the high-sensitivity Pilatus3-100K photon-counting area detector or by scattering at $\sim$ 90 ∘ from a graphite analyzer crystal into a photo-diode. Rotating the analyzer crystal about the scattered wave-vector provides a measurement of the polarization of the diffracted signal.

III Results and Discussion

III.1 Structural properties of UN films

Figure 1 shows the variation of the lattice parameters in the UN film as a function of temperature measuring higher-order Bragg reflections. The values given by Marples (1970) Marples (1970), measured from a polycrystalline sample, are shown as inverted triangles. From this it can be seen that our UN films are slightly expanded by $\sim$ 0.004 Å ($\sim$7 $\times$ 10*-4* in terms of strain) in the growth direction due to the interaction with the buffer and substrate. The lattice linear expansion coefficient from 100 - 300 K is approximately the same as that reported in Ref. Marples (1970), but from the in-plane lattice components we can see that the film is under tensile strain of $\sim$ +20 $\times$ 10*-6*, where the growth axis is larger than the mean of the in-plane parameters. Moreover, this strain increases with temperature, as the expansion of the sapphire substrate (indicated by dashed lines in the figure) is smaller than that of UN.

A further point to make here is to note the expansion of the lattice below the AF ordering temperature (TN). This will be considered more carefully below, but it represents an important measure of the interaction of the UN lattice with the magnetic (electronic) components.

Figure 2 focuses on the expansion in the UN lattice when the material orders. We find that our film has TN = 45.8(3) K, which is lower than the 52-55 K region found in bulk samples Troć et al. (2016), and is not surprising given the effect of the strain induced by the substrate. This figure includes data from previous studies Marples (1970); Marples et al. (1975); Doorn and Plessis (1977); Shrestha et al. (2017). Apart from Marples Marples (1970), all studies were performed on single crystals, although Refs. Doorn and Plessis (1977); Shrestha et al. (2017) used strain-gauge techniques, not x-rays. What is surprising about this figure is that the expansion of the lattice appears to depend on the sample form, and the magnitude is thus not a true bulk property, as it varies by almost a factor of three between different samples. Our results give a lower value, similar to that derived from a polycrystalline sample as measured by Marples Marples (1970). This is particularly interesting, as the study by Shrestha et al. Shrestha et al. (2017) shows that this expansion may be partially suppressed by the application of a modest magnetic field (¡ 30 T), although there seems no obvious explanation for this in the AF state. Magnetic fields of $\sim$ 60 T are required Troć et al. (2016); Shrestha et al. (2017) to disrupt the AF order of UN.

We now come to the question of a lattice distortion at TN. Curry was the first to report the AF structure of UN in 1965 Curry (1965); the structure consists of ferromagnetic sheets of uranium moments arranged in a simple $+$ $-$ orientation along the propagating axis, which, in the single-k form, may be any one of the cube axes \hkl¡100¿. The moments are parallel to the propagation direction. This immediately gives three possible domains (neglecting time reversal), and the symmetry is tetragonal, i.e. one would expect a distortion at TN so that $c$ (parallel to the propagation direction) is no longer equal to $a$ and $b$ (perpendicular to the propagation direction). However, the possibility of a so-called 3k structure, in which all domains exist in one unit cell, cannot be distinguished by the intensities of the reflections, and this 3k configuration has cubic symmetry. Rossat-Mignod et al. Rossat-Mignod et al. (1980) were the first to test this on UN and concluded that UN was indeed a 1k system, at least under the application of uniaxial stress. They stated that with uniaxial stress UN became tetragonal with $c/a$ >1. Marples et al. Marples et al. (1975) looked specifically with x-ray diffraction for the expected distortion and reported that $c/a$ = 0.99935(3) at the lowest temperature, i.e. the resulting strain $2(c-a)/(c+a)$ = – 6.5 $\times$ 10*-4*. Note this is the opposite sign to that suggested in Ref. Rossat-Mignod et al. (1980). A distortion implies that different $d$-spaces will be detected; for example, in the case of the \hkl(0 0 10) reflection that the $d$-space for \hkl(0 0 10) will be different from that for \hkl(10 0 0) and \hkl(0 10 0) reflections. A subsequent study, also on a single crystal, by Knott et al. Knott et al. (1980), found a smaller broadening of the full-width at half maximum (FWHM) than reported in Ref. Marples et al. (1975) and concluded that the distortion, if present, was smaller than reported in Marples et al. (1975). There is, therefore, some doubt over the existence of such a tetragonal distortion.

Synchrotron x-rays have the advantage over laboratory source x-rays that there is only one single wavelength and not a mixture (e.g. Cu K$\alpha_{1}$ and Cu K$\alpha_{2}$) in the beam, so we have used this to lower the limit found by samples to a possible strain of $\sim$ 2 $\times$ 10*-4*, as shown in Figure 3. Unfortunately, Marples et al. Marples et al. (1975) do not show their raw data of the diffraction profiles, which are simulated in our figure. However, they do show the broadening of the FWHM of their peaks, which they then analyze in terms of a distortion.

However, broadening of the peaks can also be a result of changing strain. This is shown dramatically in Figure 4, where we show what happens to the FWHM of the \hkl(0 0 10) and \hkl(555) reflections from the film. The strain along the $c$ axis (growth direction) of the film increases, whereas the corresponding in-plane strain actually compensates by decreasing. Clearly, this is a complicated effect, driven by the film-substrate interaction, but it gives no support to the idea of a tetragonal distortion in UN. If that were the case the \hkl(555) reflection should not change its FWHM, as all $d$-spaces for this reflection are the same whether a tetragonal distortion occurs or not. Thus, the \hkl(555) reflection could change its position, but should not broaden; instead it actually narrows its FWHM with decreasing temperature below TN.

The combination of Figs. 2 - 4 suggests that the tetragonal distortion in UN may not be present without the external perturbation of uniaxial stress, and that strain effects in the different samples are more important. It is possible, therefore, that the true state of the magnetic configuration is 3k, where a tetragonal distortion would not be expected. The only unique way to distinguish these two possibilities is by analysis of the polarization of the spin waves, a complicated neutron inelastic experiment performed only so far for UO2 Blackburn et al. (2005) and USb Magnani et al. (2010), both 3k systems.

III.2 Magnetic Scattering from UN and U2N3 films

III.2.1 UN

One of the possibilities with the UN epitaxial film was that the strain in the lattice because of the interaction with the substrate might induce a single magnetic domain to be observed. However, below TN eleven different magnetic reflections (all related by the known magnetic wave-vector of q = \hkl¡001¿) were observed when the energy was changed to the U $M_{4}$ edge, and no absences were found. For example in a 1k configuration, the \hkl(110) belongs to a $c$ domain (corresponding to the propagation direction) along \hkl[001], whereas the \hkl(101) reflection corresponds to a $b$ domain, and the \hkl(011) to the $a$ domain. All were present. Thus the hope that a change in the population of the 1k domains might be induced by the substrate-film interaction inducing a slightly “orthorhombic” UN (as noted in Fig. 1) was not fulfilled. On the other hand, if the true configuration is 3k, all such reflections would be present. However, this observation is not proof of a 3k AF state.

The exact intensity in resonant x-scattering (RXS) is complicated, and not related directly to the value of the magnetic moment Hill and Mcmorrow (1996). The observed intensities depend greatly on the large absorption, which at the U $M_{4}$ energy (3.726 keV) in UO2 can reach values of $\sim$ 5 $\times$ 104 cm*-1* Cross et al. (1998) corresponding to $f^{\prime\prime}$ (the imaginary part of the structure factor) reaching $\sim$ 70 electrons. In UN this corresponds to a 1/e attenuation length of $\sim$ 150 nm. Some of the beam will pass through the 70 nm film of UN at this energy, but the absorption will depend on the precise geometry and is a difficult correction to make. It is noteworthy that (so far) no report relating intensities of magnetic reflections measured at the U $M_{4}$ edge has been published. These arguments apply also to U2N3, and will not allow the magnetic structure to be determined in that material with this RXS technique. Such an investigation with RXS was reported by Watson et al. (1998) Watson et al. (1998), but at the $L_{3}$ edge of Nd, where the energy is higher than that at the U $M_{4}$ edge, and the resonant absorption (i.e. $f^{\prime\prime}$) much smaller.

Figure 5 shows the variation of the intensities of magnetic reflections from the two materials as a function of temperature. The reflections are \hkl(001) for UN and \hkl(003) for U2N3. Previous work on bulk UN Holden et al. (1982) has given a value of $\beta$ = 0.31(3), and early work on a similar bulk rocksalt uranium compound USb Lander et al. (1978) gave a value of 0.32(2). The value determined here, 0.53(5), for UN appears significantly higher. However, there is evidence that critical exponents from thin films are not necessarily the same as those determined from bulk samples. A good example is our recent work on UO2 Bao et al. (2013), where the values for thin films range considerably in value, and indicate a 2nd-order phase transition, whereas bulk UO2 has a 1st-order transition at TN. The value found for U2N3 is consistent with a simple mean-field model for the transition.

Figure 6 shows the lattice parameters extracted from longitudinal scans of the specular reflections, as a function of temperature, together with their relative widths $\Delta d/d$. All the reflections taken at 15 keV are in good agreement with one another (as they should be), but lattice parameters measured at the resonant energy appear to be greater. This is due to refraction effects and has been known for many years James (1965). Normally, these effects are small, but we have an unusual case of using relatively long wavelengths x-rays, $\lambda$ = 3.327 Å, at the U $M_{4}$ resonant edge, and the electron density per unit cell, $n_{e}$ = 3.41 electrons/Å3, is high because of the uranium.

For specular type reflections Greenberg (1989) Greenberg (1989) has shown that Bragg’s law can be re-written as

[TABLE]

where $\delta$ is the correction to the refractive index defined as $n=1-\delta$, assuming $n=1$ in vacuum. This may be readily transformed in the simple case of a specular reflection from sets of planes perpendicular to the growth direction to note that the change in the effective $c$ lattice parameter $\Delta c$ is given by

[TABLE]

$\delta=r_{0}\lambda^{2}n_{e}/(2\pi)$, where $r_{0}$ is the classical electron radius, $r_{0}$ = 2.82 $\times$ 10*-5* Å, giving a value of $\delta$ = 1.7 $\times$ 10*-4*. Normally these values seldom exceed a few parts in 10*-5*. Here we have also changed $\theta$ to $\theta^{\prime}$ to reflect the fact that we are very close to, but not actually at, the specular condition. This is because of the miscut in the substrate. $\theta^{\prime}$ represents the angle between the beam and the surface of the film.

Such effects are much more noticeable for low-angle reflections as the effect is proportional to $1/sin^{2}\theta$, which implies [given the almost linear relationship for the low-angle reflections between the reflections \hkl(00L) and $sin\theta$ for the Bragg reflections] that the effect for the \hkl(001) reflection is $\sim$ 4 times more pronounced than for the \hkl(002). The dashed lines in Fig. 6 show the values derived by assuming that the 15 keV data give the true value, and the refractive index correction has a value of $\delta$ = 2.0 $\times$ 10*-4*. Given the approximations made with the miscut and treatment of absorption at the lower energy, the agreement with calculated value (1.7 $\times$ 10*-4*) is satisfactory.

By comparison for the \hkl(0 0 10) reflection using 15 keV x-rays, the correction $\Delta c/c$ $\sim$ 0.15 $\times$ 10*-4*, which is smaller than the error bars in Fig. 6.

Finally, we note in Fig. 6 an unusual effect for the magnetic \hkl(001) reflection as a function of temperature. There appears to be a steady increase in the effective $c$ lattice parameter around TN, and this is accompanied by a systematic increase in the width of the reflections $-$ signifying a decreased correlation length in the critical regime of UN as the sample is warmed through TN. This shift cannot, of course, be due to refraction, as the wavelength does not change in these measurements. A simple explanation might be that the magnetic correlations become incommensurate with the underlying lattice, but in that case two diffraction peaks would be observed. The \hkl(001) magnetic peak in UN arises from the reciprocal lattice points \hkl(000) $+q_{m}$ and \hkl(002) $-q_{m}$, where $q_{m}$ is the magnetic wave-vector, and if $\left|q_{m}\right|\neq$ 1, then two peaks would be observed, symmetric about the \hkl(001) position. There is no sign of two such peaks. Instead we have a small shift (Dq) in the parameter coupling the magnetic Bragg peak to the lattice; it is as if the magnetic correlations are connected to a lattice with a slightly different (larger) spacing.

This unusual effect has been observed previously at the U $M_{4}$ edge, Bernhoeft et al. (2004) Bernhoeft et al. (2004) with the compound USb. We shall not discuss this at length here, as Ref. Bernhoeft et al. (2004) gives a general overview of experiments on various samples, and proposes an explanation, albeit complicated, to understand this shift. Subsequent to this work in 2004, an effort was made to see whether the shifts could also be observed with neutron diffraction, where the resolution is not normally as good as with synchrotron x-rays. The successful observation with neutrons, Prokes et al. (2009) Prokeš et al. (2009), demonstrates that the effect is not related to the surface of the sample, nor is it a property unique to actinide compounds. We note that this effect is always associated with an increase in the width of the magnetic diffraction peaks, signifying a reduced correlation length. This may be seen clearly by noting that the upturn in both panels of the \hkl(001) position and relative width in Fig. 6 occur near TN.

III.2.2 U2N3

The magnetic structure of U2N3 has magnetic peaks that appear at 73.5 K (see Fig. 5) in positions in which $h$ + $k$ + $l$ = odd, i.e. they do not overlap with the charge reflections from the bcc structure where the $h$ + $k$ + $l$ = even. The magnetic wave vector is q = \hkl¡001¿. This implies that the uranium atoms related by the bcc translation have oppositely directed magnetic moments. Figure 7 shows the polarization analysis scans of three different reflections, one magnetic, and two charge, along the specular direction \hkl[001] of the U2N3 film at 10 K. Purely magnetic scattering in this configuration is $\sigma$ to $\pi$, and purely charge scattering is $\sigma$ to $\sigma$.

The bixbyite cubic structure of U2N3 also exists with transition metal ions, i.e. $\alpha$-Mn2O3. However, these materials often have a crystallographic distortions associated with the ordering, Cockayne et al. (2013) for example. Since we have not detected any distortion of the unit cell [note the symmetric shape of the \hkl(004) reflection in Fig. 7], it may be more appropriate to consider the trivalent rare-earth systems, e.g. Er2O3 and Yb2O3, as investigated by Moon et al. (1968) Moon et al. (1968).

The magnetic structure of U2N3 is similar to that found in Yb2O3 Moon et al. (1968); reference to Fig. 3 of that paper shows a complex non-collinear magnetic structure with the moments directed along their local symmetry axes. Of course, since Yb2O3 orders at 2.25 K, the exchange interactions in U2N3 and Yb2O3 are clearly different (For a start, Yb2O3 is an insulator, U2N3 is a semi-metal), so there is no reason to expect similar magnetic structures. Normally, ordering temperatures in the actinides are higher than those of isostructural compounds in the rare-earths, simply because of the larger spatial extent of the 5$f$ electrons in the actinides, and the fact that such electrons often lie close to EF, whereas the 4$f$ electrons of the rare-earths lie well below EF.

III.3 Energy dependence of scattering from UN and U2N3 films

As is well known Hill and Mcmorrow (1996), the magnetic scattering from the E1 dipole term, corresponding to the $M$ edges of uranium and dipole transitions between the filled 3$d$ core level and the partially filled U 5$f$ shell, can be represented by a complex quantity, and thus couples to the $f^{\prime\prime}$ term of the scattering factor. The imaginary part (related directly to the absorption) is large at the absorption edge. We show in Figure 8 (top 2 panels) the energy dependence for a charge \hkl(002) and magnetic \hkl(001) reflection in UN at base temperature. The \hkl(002) reflection has a standard charge profile (corresponding to the real part $f_{0}$ + $f^{\prime}$), whereas the magnetic \hkl(001) reflection has an energy dependence corresponding to the $f^{\prime\prime}$ term. The highly symmetric curve with a FWHM $\sim$ 6 eV is typical for work with thin films Bernhoeft et al. (1998), and also reflects the partial coherence of synchrotron beam at this energy. Previous experiments on I16 Bao et al. (2013) have shown that the energy width can be used to determine whether the film is ordered throughout its depth. In this case the $\sim$ 6 eV FWHM is expected, so the film is fully ordered.

A key question the experiments above have not answered is whether both uranium sites in the cubic U2N3 structure order magnetically. It would be possible to answer this if we could reliably make the absorption corrections, but this is not the case with a 200 nm film and an absorption attenuation length of about the same order of magnitude, as discussed above.

It is instructive at this stage to consider the geometric structure factors governing the charge peaks in U2N3, and these are shown in Table 1 for the first 11 reflections arranged in order of $Q$, the momentum transfer. In comparing with experiments note that the system is cubic so \hkl(hkl) values can be permuted.

There are two uranium sites in U2N3, which adopts the structure of the centro-symmetric space group # 206. Eight U1 atoms in the unit cell sit on sites with three-fold rotational inversion symmetry (C3i), and twenty-four U2 atoms sit on sites with two-fold rotational symmetry (C2). There is one adjustable positional parameter for the U2 sites with positions \hkl(x 0 14) etc., and none for the U1 atoms with positions \hkl(14 14 14) etc. X-ray Masaki et al. (1972) and neutron diffraction Tobisch and Hase (1967) have been used to determine the positional parameter for U2 and the consensus value is $x=-0.02$, which we have used in the calculations below. The presence of a finite charge intensity (i.e. $\sigma$ to $\sigma$ scattering) at the position \hkl(002) [Fig. 7] is direct proof that $x$ for U2 is not zero.

This table is simply the geometric term in the structure factor only for the uranium ions. The 48 N atoms will, of course, contribute to the total intensity, but only by a small amount, and we can neglect this.

Our initial interest in these reflections was in those of group E, where the contributions from the two different U atoms cancel. Of course, this statement is true for the spherical charge density contributions, but aspherical electron distributions, arising from quadrupoles, will stand out after the spherical part cancels. The symmetry elements of both U sites in U2N3 allow an interesting phenomenon called anisotropic tensor scattering (ATS) Collins et al. (2001); Kokubun and Dmitrienko (2012) to be observed. The symmetry implies that aspherical (quadrupolar) electron distribution may exist around both U sites. A consequence of the reduced symmetry (compared to the very high symmetry observed, for example, in fcc UN) is that the local configuration around each U atom may not have an inversion center. This may be seen from Fig. 1 of Ref. Moon et al. (1968). Instead of an eightfold coordination of nitrogen about each actinide ion, there are only 6 for both the C3i and C2 sites. The positions of the nitrogen are also marked in Ref. Troc (1975), Fig. 7, and these figures also show the non-centrosymmetric local coordination around each U atom. Although such a lack of inversion center may be related to the physical reason the quadrupoles exist, we should emphasize that the symmetry elements alone determine whether this phenomenon can be observed.

The full scattering factor may be written

[TABLE]

where the first two terms are real, and the last term is imaginary. The last two terms are zero unless the x-ray wavelength is near an absorption edge. Normally, if the environment of the atom is highly symmetric, the energy dependence of the charge scattering will resemble the well-known dispersive shape, as shown in Fig. 8(a). However, if the spatial dependence of the electronic distribution has an aspherical contribution from the quadrupoles, then, with the energy close to an absorption edge this aspherical distribution will couple to $f^{\prime\prime}$. Of course, the spherically symmetric part will always be present, so the much smaller asymmetric part cannot be observed unless the spherically symmetric part cancels. U2N3 gives a good illustration of this, as shown in the lower part of Fig. 8.

Figure 8(c) shows the energy dependence of various charge reflections allowed in the bcc structure of U2N3. With reference to the groups listed in Table 1, we see that groups B, C, and E sense the imaginary term $f^{\prime\prime}$, whereas group D has the real part, $f_{0}+f^{\prime}$, energy dependence. Group A reflections have all contributions in phase and are not sensitive to the aspherical distribution. Group B are forbidden reflections, group C would be zero if $x=0$ for the U2 atom, and are thus weak, and group E have the spherical contribution from the two U sites cancelling, and are thus also weak. The scattering from the nitrogen atoms, present in all reflections, except group B, is weak compared with any scattering from uranium, so allows the ATS contribution to be also observed in groups C and E.

Previously, ATS scattering has mainly been observed at transition metal $K$ edges Collins et al. (2001); Kokubun and Dmitrienko (2012). However, disentangling the physics from such $K$-edge measurements is difficult, as the $K$ edge corresponds to 1$s$ to n$p$ dipole transitions, where n is the first partially filled $p$ shell, and higher-order transitions, (for example, the quadrupole transition is 1$s$ to n$d$) can contribute. In our case, we know that the transitions are dipole in nature and that the aspherical part of the electron density involves the 5$f$ electrons, because we observe the effect at the U $M_{4}$ edge. To our knowledge no such comparable observation involving 5$f$ electrons has been reported previously. To verify that this scattering is truly ATS we have performed an azimuthal scan (not shown) on the \hkl(002) reflection, and we have also shown that the ATS scattering of all reflections is independent of temperature. The magnetically-driven ordering of quadrupoles, such as is found in UO2 and NpO2 Santini et al. (2009), would have a different azimuthal and energy dependence, and are dependent on temperature, with no signal for T >TN.

Table 1 (final column) shows that there are contributions of the ATS from all reflections from the U2 atom, but groups C and E have no contribution from the U1 atom. Since we see ATS scattering from group C and E reflections, the aspherical distribution must be present around the U2 site. Although we cannot exclude its presence around U1, the fact that the \hkl(022) (group E) and \hkl(013) (group B) contributions, in Fig. 8 (c), are of the same magnitude, suggests that any U1 aspherical contribution is very small, as the \hkl(013) has contributions from both U1 and U2, whereas the \hkl(022) has contributions from only U2. These two reflections have a Lorentz factor ($1/sin2\theta$) of $\sim$ 1.2, and are close in $\left|Q\right|$. The Lorentz factor for the \hkl(002) reflection is 1.71 so it should be stronger than reflections at higher Q. The overall Q-dependence for intensities from this dipole transition is still subject of discussion Hill and Mcmorrow (1996); Watson et al. (1998).

IV Conclusions

IV.1 UN

Early work on UN assumed that the electronic configuration was 5$f^{2}$ and that the 5$f$ electrons were localized. Using well-known crystal-field theory, many of the experimental results could be explained on this basis. However, the first neutron inelastic scattering experiments on UN in 1974 failed to find any distinct crystal-field levels Wedgwood (1974), and no evidence has been found for such levels in more recent experiments Aczel et al. (2012); Lin et al. (2014), so that this theory does not seem immediately relevant. We assume that the crystal-field levels are heavily damped by the interactions between the 5$f$ and conduction-electron states. The advent of band theory changed the perspective on even some of the earlier experimental results, in that the orbital moment could be incorporated into such theories Brooks and Kelly (1983). Two important experiments, both on single crystals, added further weight to the idea that the 5$f$ electrons are itinerant in UN, firstly, the neutron inelastic scattering Holden et al. (1984), and, secondly, the measurement of angular resolved photoemission spectroscopy Fujimori et al. (2012, 2016).

Although, some properties of UN might still be described with localized 5$f$ states, giving rise to the idea of duality in UN Troć et al. (2016), the weight of evidence points to the best approach being one with band (itinerant) 5$f$ electrons. The theoretical calculations mentioned earlier Yin et al. (2011); Szpunar and Szpunar (2014) both use such assumptions, with the 5$f$ states numbering between two and three 5$f$ electrons. On the other hand, these calculations do not reproduce the correct (as measured by experiment) AF magnetic moment in UN, and no effort that we are aware of has been made to calculate the ground-state antiferromagnetic state and associated moment - this being an excellent test as to whether the assumed electronic structure is a true representation of the material. When such calculations are made, we can see which of the 1k or 3k magnetic configurations is the more stable state. More complicated is to replicate the expansion of the unit cell below TN, and the effects of the magnetic fields on the system Troć et al. (2016); Shrestha et al. (2017), which depend crucially on the AF ground state. These large effects with applied magnetic field Gorbunov et al. (2019) reflect a strong coupling between spin, electronic, and lattice degrees of freedom.

IV.2 U2N3

Although this material is closely linked to UN, very few investigations of its electronic structure have been reported. The crystal structure and its synthesis are well recorded. We have found that our sample orders at 73.5(2) K, with a lattice parameter of 10.80 Å. The magnetic wave-vector is q = \hkl¡001¿, which is the same as that found for Yb2O3 Moon et al. (1968). The magnetic configuration may well be non-collinear, but we cannot determine this from our measurements.

In terms of valencies on the individual atomic sites, this is a question that might be answered if we knew whether both sites ordered magnetically. We cannot answer that conclusively with the results of the x-ray experiments reported here. From the U 4$f_{7/2}$ spectra reported for U2N3 in Lawrence Bright et al. (2018) there is a shift in the weight of the spectra towards a higher oxidation state than reported for UN. If we assume that the majority 24 U2 sites are 3+ (i.e. approximately the same as in UN), then charge compensation (taking N3-) suggests the 8 U1 sites might well be of a higher oxidation state, perhaps even U6+. Since U6+ is soluble in water and highly reactive, this would explain why the U2N3 is more reactive (in H2O2) than either UO2 or UN, as reported in Ref. Lawrence Bright et al. (2019). U6+ would have no associated 5$f$ electrons, so would not order magnetically, nor should there be any aspherical resonant scattering from U1 as suggested by the observations in Fig.8 (c). This would be consistent with our observation that the U1 atom probably has zero (or at least small) aspherical contribution to the ATS scattering.

Of course, such counting of charges is certainly too simple an approach in a material that is certainly a semimetal, and we welcome some theoretical interest given that U2N3 is always found in conjunction with (and especially at the surface of) UN.

Similarly, a theoretical investigation should be able to throw light on the ATS scattering reported for U2N3 shown in Fig. 8 and believed to be associated only with the U2 atom. For example, it seems probable that this is related to the hybridization between the U 5$f$ states and the N 2$p$ states and directly represents 5$f$ covalency. Such effects might be a widespread property of actinide compounds, but for its observation requires the special conditions afforded by forbidden reflections in the bixbyite structure Kokubun and Dmitrienko (2012), which exists for U2N3, but not for UN.

Acknowledgements

We acknowledge Diamond Light Source for time on Beamline I16 under Proposal 20776 and funding from EPSRC grant 1652612.