Sylvain Le Tonquesse

Mathieu Pasturel

Valérie Demange

Akhil Tayal

Pier Lorenzo Solari

Carmelo Prestipino
email: carmelo.prestipino@univ-rennes1.fr

Valerie Demange

X-ray absorption study on the origin of the deviation from Vegard's law for U(Al1-xGex)3 solid solution

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Similarly to silicon, germanium added to the nuclear material UAl 3 was shown to efficiently block the formation of the undesired UAl 4 during the fabrication process or under irradiation by the reaction of the former with the aluminium matrix in fuel plates or irradiation targets [START_REF] Thurber | Development of silicon-modified 48 wt.% ual alloys for aluminum plate-type fuel elements[END_REF]. Moreover, it also has 5 a protective role against the formation of the detrimental ternaries UMo 2 Al 20 and U 6 Mo 4 Al 43 in γ-U(Mo)/Al(Ge) diffusion couples simulating prospective fuel plates based on these components [START_REF] Perez | Phase development in a u-7wt. % mo vs. al-7wt. % ge diffusion couple[END_REF]. A deep knowledge of the local and long range order of the U(Al,Ge) 3 solid solution is thus required to simulate as precisely as possible the reactivity and irradiation behavior of this fuel. But opposite to the case of U(Al,Si) 3 for which numerous data are available in literature [START_REF] Rabin | Thermodynamic modeling of al-u-x (x= si, zr)[END_REF]4,[START_REF] Ryu | Heats of formation of (u, mo) al3 and u (al, si) 3[END_REF], scarce reports are available for the U(Al,Ge) 3 solid solution [START_REF] Tyunis | Valence instability of uranium in U(Al1-x Gex)3[END_REF]7].

The binary intermetallic compounds UAl 3 and UGe 3 crystallize in the CuAu 3 structure-type (space-group Pm 3m) where U is located at the 1c and Al or Ge at the 3a Wyckoff positions as shown in Fig. 1 (a). Because they crystallize in the same structure-type with relatively close lattice parameters and similar covalent radii 1 , (r Al = 1.21 Åand r Ge = 1.20 Å [START_REF] Cordero | Covalent radii revisited[END_REF]), the existence of a total solid-solution as reported by Tyunis et al. [START_REF] Tyunis | Valence instability of uranium in U(Al1-x Gex)3[END_REF] and later on by Moussa et al. [7] is not surprising. Indeed, similar pseudo-binary systems USn 3 -UGa 3 , USn 3 -UAl 3 [11] and UAl 2 -UCo 2 [12] are solid solutions with a complete miscibility and present a conventional evolution of lattice parameters.

For solid solutions, lattice parameters are expected to vary linearly with the composition between the extremities, in accordance with the empirical Vegard's law [13]. The observance of such law agrees with the additivity of covalent radii, as earlier suggested by Pauling and Huggins [START_REF] Pauling | Covalent Radii of Atoms and Interatomic Distances in Crystals containing Electron-Pair Bonds[END_REF], implying that atomic volumes remain constant independently of the extent of the mixing. However this assumption has been experimentally proven only much later with the development of local range techniques such as EXAFS, because more common Bragg diffraction techniques are based on the coherent scattering generated by long range order, i.e. the information obtained are averaged along many unit cells.

For instance, in the seminal work of Mikkelsen and Boyce [15], for the first time a solid solution obeying to the Vegard's law, namely zinc-blende In x Ga 1-x As (S.G. F 43m), has been investigated as a function of the composition by EX-AFS spectroscopy and diffraction. Bond distances derived from diffraction go from 2.45 Å for GaAs to 2.62 Å for InAs. However, EXAFS detects that lattice 1 Metallic radii more commonly used in the intermetallics community are also similar, although the actual metallic radius of Ge is less clearly defined (r metal Al = 1.432 Å and r metal Ge = 1.378 Å for Teatum [START_REF] Teatum | Compilation of calculated data useful in predicting metallurgical behavior of the elements in binary alloy systems[END_REF] and r metal Ge = 1.43 Å for Batsanov [START_REF] Batsanov | Metallic radii of nonmetals[END_REF] induced strain affects linearly the bond only up to -2% and +1%, for As-In and As-Ga bonds and consequently two almost constant but clearly different distances are present for all solid solution compositions.

However, although the similitude with the other U pseudo-binary system, both Tyunis et al. and Moussa et al. [START_REF] Tyunis | Valence instability of uranium in U(Al1-x Gex)3[END_REF]7] reported that the lattice parameter of the U(Al 1 -x Ge x ) 3 solid-solution deviates negatively from linearity and thus violate the Vegard's law. A valence instability of uranium or relative size effects of the p-element were evoked, respectively, to explain this observation.

In the present study, a local structure examination of U(Al 1-x Ge x ) 3 intermetallics with x=0, 0.25, 0.5, 0.75 and 1 using EXAFS and XANES measurements at U L 3 and Ge K-edges has been performed to better understand the structural origin of such deviation and improve the comprehension of chemical bonding in this potential nuclear material.

Experimental

Polycrystalline ingots with nominal compositions U(Al 1-x Ge x ) 3 , with x = 0, 0.25, 0.5, 0.75 and 1, were prepared by arc melting stoichiometric amounts of elemental components (all purities ≥ 99.5 %). The sample were turned upside down and remelted twice to ensure homogenisation. A subsequent annealing was performed at 973 K for two weeks after enclosing the samples in evacuated silica tubes. Resulting samples have been investigated by X-ray diffraction (XRD) using a Bruker D8 Advance diffractometer (θ -2θ Bragg-Brentano geometry, monochromatized Cu Kα 1 , λ = 1.5406 Å) on powders obtained by manually grinding the ingots in an agate mortar. XRD data were modelled with the help of the FullProf code [16]. Scanning electron microscopy was performed using a JEOL JSM 7100 F microscope while electron diffraction and elemental analysis (Energy dispersive spectroscopy,EDS) were performed by transmission electron microscopy on a JEOL 2100 LaB 6 instrument operating at 200kV. To select the smallest particles, powders were suspended in ethanol before deposition on copper grids. Then the grids were kept for a few minutes in air to let the solvent evaporate.

For X-ray absorption spectroscopy, the U(Al 1-x Ge x ) 3 powders (≈ 10 mg) were diluted with cellulose in order to achieve a maximum absorbance of ≈ 2, pressed into pellets and then doubled confined in Kapton tape and polyethylene foil. The measurements were carried out on MARS beamline at SOLEIL synchrotron (Saint-Aubin, France) [17]. The X-ray beam energy for measurement at the U L3-edge was selected and scanned using a sagittal focusing Si(220) crystals monochromator and harmonic rejection has been performed by the use of a couple of Pt coated mirrors with an angle of 3.1 mrad. For the measurement at the Ge K-edge, sagittal focusing Si(111) crystals monochromator and Pt mirrors angle of 2.2 mrad were used instead. Extended X-ray Absorption Fine Structure (EXAFS) data have been collected at Ge K edge (11403 eV) and at U L 3 edge (17166 eV) in transmission mode using specific intensity monitors with silicon diodes. Energy calibration has been monitored by measuring simultaneously the absorption of reference sample with residual photons (Y and Ge foils for U L 3 and Ge K edge, respectively). High Energy Resolution Fluorescence Detection (HERFD) X-ray absorption has been also collected on Ge K edge and at U L 3 edge using a spectrometer equipped with a Si(660) or Ge(777) secondary monochomator crystal selecting Ge K-L 3 (9886 eV) and U L 3 -M 5 (13615 eV) signals, respectively. For every compositions U(Al 1-x Ge x ) 3 (x= 0, 0.25, 0.5, 0.75 and 1) fine structures µ(E) after the U L 3 and Ge K edges were measured and converted into χ(R) by Fourier transform using 1, 2 and 3 k-weight and Hanning window with dk = 1. The k-range used for the Fourier transform where 3.5 -15 Å-1 both for the Ge K and U L 3 edges. The fitting procedure was realized simultaneously on the χ(R) of both edges over a range of 1.6 -3.6 Å for Ge K and 1.6 -4 Å for U L 3 . The coordination spheres around U and Ge as determined from XRD was used to calculate phase and amplitudes employed as initial model for fitting procedure. Reduction and modelling of the data were realized using the IFEFFIT code as implemented in the Athena and Artemis softwares [18]. Finally XANES simulation has been performed using FEFF8.4 code in the muffin-tin real space : full multiple scattering approach. Self-consistent poten-tial calculations were performed using Hedin-Lundqvist self-energy in a cluster containing around 100 atoms

Results

Samples of composition U(Al 1-x Ge x ) 3 with x=0, 0.25, 0.5, 0.75 and 1 were synthesized and analyzed by powder diffraction (PD). The patterns of all the compositions are indexed with the cubic CuAu 3 structure-type (Pm 3m) and only a small amount of UC impurity (Fm 3m) could be detected on the pattern of UAl 3 (Figure 1b). Le Bail refinements of the UAl 3 and UGe 3 patterns converges to lattice parameters of 4.2676(1) Å and 4.2078(2) Å, respectively, in good agreement with literature data [19,20]. By assuming the solid-solution to follow the Vegard's law, compounds with intermediate composition are expected to have a lattice parameter following the linear relation:

a = -0.0199 • x + 4.2676 Å. (1)
However, experimental lattice parameters found for x = 0.25, 0.5 and 0.75 where a is the lattice parameter obtained from powder diffraction. For U-U contribution, the U-U distances were constrained to the value of a and only the mean square relative displacements (σ 2 ) were allowed to vary. For the other contributions, the degenerates (N ) of Al and Ge were constrained according to the stoichiometry, i.e. using a binomial distribution. The refined structural parameters were the NN distances along with their σ 2 . Fig. 4 shows the quality of the fits within the R range corresponding to the first coordination spheres (1.0 -3.75 Å) while Table 1 summarizes the values of the fitted parameters obtained after the refinements. It is worth of noticing, that although the relatively high number of parameters for the fit of a single signal, the correlation between parameters remain rather limited due to the fact that both edges are fitted simultaneously implying that structural parameters remain the same for both signals. The χ(R) of UAl 3 and UGe 3 , shown in Figure 4

U-Al/Ge = U-Ge • x + U-Al • (1 -x) (2)
With increasing the Ge-content in UAl 3 , the U-Al distance was found to shorten from 3.022(4) Å in UAl 3 to 2.98(2) Å in UAl 0.75 Ge 2.25 . As mentioned before, this is the typical behavior of conventional solid solutions where only a light shortening of distance, due the increasing crystal strain, is expected. As a consequence the average U-Al/Ge distance, strongly deviates from the 205 linear evolution expected from the Vegard law. Remarkably , the trend of the deviation calculated from EXAFS data is in good agreement with the deviation measured by XRD since both are negative and with the maximum deviation for the UAl 1.5 Ge 1.5 and UAl 0.75 Ge 2.25 compositions. However, the extent of such deviation is systematically higher for EXAFS data in respect to XRD, suggesting the possible existence of medium range distortion that tends to accommodate the bond distance differences. In this context the deviation from Vegard's law is directly caused by the abnormal shortening of the U-Ge distance with increasing Al-content.

In order to investigate the origin of such Ge-U bond behavior, the evolution of XANES spectra as a function by of the composition was measured by high energy resolution fluorescence detection (HERFD) on the two U L 3 and Ge K edges. HERFD is an acquisition mode for X-ray absorption in which the fluorescence signal is recorded with crystal analyzer with an energy resolution higher than the core hole [START_REF]X-ray absorption and X-ray emission spectroscopy: theory and applications[END_REF]. Such setup has the major effect to keep constant the exchange energy during the photon in photon out process, with the consequence of replacing in the spectral broadening the contribution of the photon excited core level with the one related to the fluorescence emission. In the present case, it enabled to replace the intrinsic resolution for U spectra from 8.16 eV to 3.94 eV and for Ge from 2.36 eV to 0.76 eV.

The HERFD spectra for the studied compositions for both U L 3 edge and Ge K edge are shown in Fig. 6. The U L 3 edge is dominated by the dipole allowed transition 2p →6d commonly called white line at around 17175 eV.

The XANES spectroscopy of Uranium based intermetallics, has been widely investigated by several groups [START_REF] Booth | Multiconfigurational nature of 5f orbitals in uranium and plutonium intermetallics[END_REF][START_REF] Booth | Delocalization and occupancy effects of 5f orbitals in plutonium intermetallics using L3-edge resonant X-ray emission spectroscopy[END_REF][START_REF] Booth | Probing $5f$-state configurations in ${\mathrm{URu}} {2}{\mathrm{Si}} {2}$ with U ${L} {\text{III}}$edge resonant x-ray emission spectroscopy[END_REF][START_REF] Kvashnina | Resonant x-ray spectroscopy of uranium intermetallics at the ${M} {4,5}$ edges of uranium[END_REF], determining a good correlation between the position of the white line peak with f levels population. In this context the progressive blue shift of the white line and edge position as the Al concentration increases should be interpreted as a limited depopulation (oxidation) of U 5f levels of around 0.2 e -taking in account a shift around 2 eV [START_REF] Booth | Multiconfigurational nature of 5f orbitals in uranium and plutonium intermetallics[END_REF].

The observation is in agreement with the work of Tyunis et al. The depopulation of the U-5f orbitals with increasing aluminum content is supported by specific heat measurements (fig. 7). Despite some noise at the lowest temperatures in the signal of UAl 3 due to the non-congruent melting of this phase [START_REF] Pauling | Covalent Radii of Atoms and Interatomic Distances in Crystals containing Electron-Pair Bonds[END_REF]7] inducing the appeareance of porosity and cracks after annealing, the specific heat follows up to about 10 K the expected law for a regular metal:

C p T = γ + βT 2 (3)
where γ is the Sommerfeld coefficient and β the Debye component of the specific heat.

The specific heat of UAl 3 agrees with previous reports, showing an almost flat variation of C p /T = f(T 2 ) below 100 K 2 (β = 0.118 mJ mol -1 K -4 ) and a Sommerfeld coefficient of 43 mJ mol -1 K -2 (41, 41.6, 43.2 and 47 mJ mol -1 K -2 for Scheidt [11], van Maaren [START_REF] Van Maaren | High electronic specific heat of some cubic ux3 intermetallic compounds[END_REF], Cornelius [START_REF] Cornelius | Electronic properties of u x 3 (x= ga, al, and sn) compounds in high magnetic fields: Transport, specific heat, magneti-365 zation, and quantum oscillations[END_REF] and Aoki [START_REF] Aoki | Fermi surface properties of the enhanced pauli paramagnet ual 3[END_REF] et al., respectively).

The values obtained for UGe 3 (β = 0.49 mJ mol -1 K -4 , γ = 20.2 mJ mol -1 K -2 ) are also in line with literature data (β =0.455 mJ mol -1 K -4 [28] -γ = 20.4 [START_REF] Van Maaren | High electronic specific heat of some cubic ux3 intermetallic compounds[END_REF][START_REF] Pikul | Low-temperature specific heat of uranium germanides[END_REF], 20.5 mJ mol -1 K -2 [11]). With the exception of UAl 2.25 Ge 0.75 , β increases regularly with the Ge-content, in agreement with the heavier molar mass of this element. γ follows an opposite increase with the Al-content along the solid solution. Such an increase highlight an enhanced contribution of the electrons to the specific heat which is usually attributed to a higher delocalization of the U-5f electrons, in agreement with the spectroscopic observations.

The anomalous behavior of UAl 2.25 Ge 0.75 , confirmed by C p measurements on several pieces with this composition, remains unexplained yet.

Conclusion

In the present work the origin of the deviation from the Vegard's law for HERFD XANES and specific heat measurements show that such unexpected decrease of bond length is related to the depopulation of the U 5f levels (around 0.2 e -) suggesting that the deviation from Vegard's law is caused by valence instability in agreement with the work by Tyunis et al. [START_REF] Tyunis | Valence instability of uranium in U(Al1-x Gex)3[END_REF].

280

Finally, such valence instability, in accordance with XANES simulations, seem to be related to the hybridization of Ge 4p levels with U 6d levels.

were 4 .

 4 2511(2), 4.2271(1) and 4.2132(2) Å, respectively, which indeed correspond to a negative deviation from (1) reaching a maximum of ≈ -0.007 Å in the case of UAl 0.75 Ge 2.25 .Deviation from the Vegard's law is not uncommon and were reported for many classes of materials such as oxides, ionic salts or intermetallic. It can originate either from structural effects due to not totally random site occupancies and distortions, such as long range ordering or clustering[21] or from an actual variation of involved bond distances, implying a change in the electron configuration of the material.Long range ordering phenomena can be detected by diffraction experiments, if their coherence length is sufficient to generate superstructure peaks. For example, in the closely related UAl 3 -USi 3 system, crystallizing in the same CuAu 3 structure-type (Pm 3m), Meshi et al. observed the appearance of satellite

Figure 1 :

 1 Figure 1: (Color online) (a) Structure of UAl 3 and UGe 3 crystallizing in the AuCu 3 structuretype (Pm 3m); (b) powder XRD patterns of the U(Al 1-x Gex) 3 samples with x=0, 0.25, 0.5, 0.75 and 1. The reflections marked with black dots (•) on the UAl 3 pattern correspond to the UC impurity

Figure 2 :

 2 Figure 2: (Color online) Electron diffraction patterns of UAl 1.5 Ge 1.5 (top) and UAl 0.75 Ge 2.25 (bottom) along the [100] (right), [110] (middle) and [111] (left) zone axis confirming the absence of satellite reflections.

Figure 3 :

 3 Figure 3: (Color Online) Left and central parts: experimental EXAFS data for U(Al 1-x Gex) 3 solid solution at U L 3 (green curves) and Ge K (blue curves) edges. χ(k)k 2 and imaginary part and magnitude Fourier transform of few selected compositions. Right side: Coordination spheres around U and Ge used as initial model for EXAFS refinements. (a) U-atom coordinated with 12 Al/Ge forming a regular cuboctahedron and 6 more distant U forming a regular octahedra (b) Ge-atom coordinated with 12 Al/Ge forming a regular cuboctahedron

  (a) and (e), could be fitted using a single NN distance of 3.022 Å for U-Al and 2.975 Å for U-Ge in good agreement with XRD distances (3.017 Å and 2.975 Å , respectively) proving the adequacy of the approximation of considering the calculation of phase and amplitude for U L 3 sufficiently precise to be directly compared with powder diffraction. It is worth to notice that for the alloys, no satisfactory fit could be obtained constraining U-Al and U-Ge to the same bond length. Appropriate fits (Fig.4 (b)-(d)), could be achieved only using two different bond lengths as expected from the previous solid solution works on local structure.

Figure 4 :

 4 Figure 4: (Color Online) Fits of the χ(R) for all the compositions studied in the 1.0 to 3.75 Å range corresponding to the first coordination sphere of U-and Ge-atoms in the U(Al 1-x Gex) 3 solid solution. χ(R) of the Ge K and U L 3 edges are plotted in blue and green, respectively. Results of the fits are plotted in red.

Fig. 5

 5 Fig. 5 shows the evolution of the U-Ge and U-Al distances as a function of the concentration x of Ge in U(Al 1-x Ge x ) 3 along with the average U-(Al/Ge) distance determined by XRD and by a weighted average of the distances obtained by EXAFS analysis, calculated with:

200

  On the other hand, the U-Ge distance, in discrepancy with crystallography prediction where the distance is supposed to increase due to the bigger cell parameter of UAl 3 , decreases with increasing Al concentration from 2.969 Å in UGe 3 to 2.941 Å in UAl 2.25 Ge 0.75 ( ≈ -1 % shortening).

Figure 5 :

 5 Figure 5: (Color Online) Evolution of the U-Al (blue squares), U-Ge (red squares) and average U-Al/Ge distances (purple stars) obtained from EXAFS as a function of the Ge concentration x in U(Al 1-x Gex) 3 intermetallics. Distances determined from XRD (green stars) and the linear relation expected from the Vegard's law (black line) are shown for comparison of the evolution obtained by the two methods.

  [START_REF] Tyunis | Valence instability of uranium in U(Al1-x Gex)3[END_REF] that have investigated the same solid solution by high resolution analysis of fluorescence emission, determining the number of 5f electrons of uranium to 2.54 for UGe 3 and 2.26 for UAl 3 (i.e. depopulation of 0.28 e -).

Figure 6 :

 6 Figure 6: (Color Online) HERFD of the U L 3 XANES for U(Al 1-x Gex) 3 with x=0, 0.25, 0.5, 0.75 and 1. The inset is a close up view showing the continuous evolution in energy of the peak edge toward higher energies with increasing Al content.

U(Al 1 -x

 1 Ge x ) 3 has been investigated experimentally by long and local range order techniques. The deviation has been proven to be based on actual bond length variation while the possible long range ordering phenomena as UAl 3 -USi 3 has been excluded by the absence of superstructures or diffuse scattering in electron diffraction. Indeed the deviation is mainly related to a decrease of U-Ge bond length induced by the increase of Al concentration in the solid solution.

Figure 7 :

 7 Figure 7: (Color Online) (a) Thermal dependence of the specific heat of UAlxGe 3-x for x = 0, 0.25, 0.5, 0.75 and 1 (symbols) and linear fit according to eq. (3); (b) Variation of the γ (left) and β (right) parameters with the chemical composition, extracted from these fits (the solid lines are guides for the eyes). The standard deviations on the values are below the thickness of the symbols.

Table 1 :

 1 Summary of the parameters obtained by combined least square refinements of χ(R) measured at the U L 3 and Ge K egdes for the U(Al 1-x Gex) 3 samples with x=0, 0.25, 0.5, 0.75 and 1. S 2 0 , ∆E 0 , N and σ 2 correspond to the passive electron reduction function, shift in the edge energy, degeneracy and mean square relative displacements, respectively. Constrained parameters are marked with *.

			UAl3	UAl2.25Ge0.75 UAl1.5Ge1.5	UAl0.75Ge2.25	UGe3
		S0 2	0.77(3)	0.75(1)	0.75(5)	0.75(2)	0.82(5)
	U-L3	∆E0 (eV)	5.6(3)	5(1)	6.2(7)	4(1)	5.2(7)
		N Al/Ge	12/0*	9/3*	6/6*	3/9*	0/12*
		S0 2	/	1.0(3)	0.81(9)	0.9(1)	0.79(8)
	Ge-K	∆E0 (eV)	/	2.99(1)	3(1)	1.9(6)	2.3(7)
		N U/Al/Ge	/	4/6/2*	4/4/4*	4/2/6*	4/0/8*
		U-Ge ( Å)	/	2.945(6)	2.946(3)	2.951(1)	2.970(1)
	bond	σ 2 ( Å2 )	/	0.007(2)	0.006(1)	0.0067(2)	0.0057(4)
	distances	U-Al ( Å)	3.022(1)	3.017(5)	3.011(3)	2.984(7)	/
		σ 2 ( Å2 )	0.0069(6)	0.008(1)	0.007(1)	0.011(4)	/
		U-U ( Å)	4.2676*	4.2511*	4.2271*	4.2132*	4.2078*
		σ 2 ( Å2 )	0.007(1)	0.04(3)	0.027(8)	0.018(5)	0.009(2)
	Agreement	χ 2	92.2	188.7	46.8	66.2	110.1
	factors	R-factor	0.021	0.039	0.023	0.007	0.016

Acknowledgements

Vincent Dorcet is acknowledged for TEM experiments performed on THEMIS platform part of the ScanMAT unit (UMS 2001, University of Rennes 1), which received a financial support from the European Union (CPER-FEDER 2007-2014).

Supplementary