The heat capacity at 298.15 K, C0
p,m(298.15K), is obtained by tting the heat capacity curve at high temperatures with a polynomial function and using an extrapolation to 298.15 K. The entropy at 298.15 K, S0
m(298.15K), is determined by numerical integration of the curve (C0 p,m/T) = f(T) between 0 and 298.15 K: Sm0(298.15K) = ∫ 298.15 0 C0 p,m(T) T dT (2.58)
2.6.2.3 Debye temperature and contributions to the heat capacity At very low temperatures, the heat capacity at constant volume, C0
V,m, is given by the sum of the contributions of the lattice vibrations and of the conduction electrons at the Fermi surface [75]. The relation between the heat capacity at constant pressure measured experimentally, C0
p,m, and the heat capacity at constant volume, CV,m0 , involves the isobaric thermal expansivity, α, and isothermal compressibility, κT, of the material:
Cp,m0 − CV,m0 =α 2V T κT
(2.59) The measurement being carried out at very low temperatures here, the thermal expansivity is negligible, so that C0
p,m ≈ CV,m0 . The energy absorption by the conduction electrons pre- dominates close to 0 K, with a term directly proportional to the temperature, whereas the lattice vibrations term in T3, derived from Debye's model, becomes the main contributor as the temperature augments further away from 0 K.
Cp,m0 (T) = γ ⋅ T + n ⋅ (12⋅ π 4⋅ R 5 ) ⋅ ( 1 θD) 3 ⋅ T3 (2.60)
where the coe cient γ characterizes the electronic speci c heat, R is the universal gas con- stant, n is the number of atoms in the formula unit, and θD is the Debye temperature characteristic for the lattice vibrations. The latter equation is valid only up to 0.1θD.
The curve C0
p,m/T as measured experimentally against T2 can hence be used to derive the electronic speci c heat contribution and Debye temperature of the material.
2.7 Thermodynamic modelling: the CALPHAD method
The CALPHAD technique provides a thermodynamic description of a system which can be used to calculate its chemical properties. It describes the equilibrium state by modelling the Gibbs energy of each phase in the system as a function of temperature, pressure, and composition. The method consists in introducing adjustable parameters into the model that are optimized to t as best as possible the experimental data available on the system (phase
diagram data, enthalpies, heat capacities, chemical potentials, etc) [76]. The thermodynamic parameters are optimized by a least-squares minimization procedure using a dedicated soft- ware (Thermocalc Version 4.1 in our case) [77, 78]. The quality of the description is then based on the agreement with the experimental data .
In this work, we used the CALPHAD method to develop a thermodynamic model for the Np-O system. A detailed description of the dierent formalisms chosen to describe each phase is given in Chapter 5. Our description is compatible with already existing models of the FUELBASE project (U-Pu-O-C [79] in particular), which is being developed since 2005. This project aims to provide a computational tool running thermodynamic calculations and assessing the behaviour of irradiated fuel materials (oxides, carbides, etc) containing ssion products, and to which minor actinides can be incorporated. It also describes the interaction between fuel and cladding material [7981].
3
Structural properties of sodium uranium
ternary oxides
3.1 Introduction
S
ince the 1960s considerable interest has existed for the characteristics of sodium ura- nium ternary oxides because of their technological importance for Sodium-cooled Fast Reactors. Their structural [8291], magnetic [90, 92], and spectroscopic properties [93100] have been reviewed extensively in the literature. The reported compounds are numerous: (Na2UO4, Na4UO5, Na2U2O7, Na6U7O24, NaUO3, Na3UO4, Na11U5O16, Na4UO4), but some of the crystal structures and chemical compositions are still questioned. A summary of the structural parameters for these phases is presented in Table 3.1.The crystal structures of α-Na2UO4, β-Na2UO4, Na4UO5, and NaUO3 are well estab- lished. The structure of Na2U2O7 was an unresolved issue until very recently. Cordfunke et al. [102] and Kovba [103] reported a monoclinic structure in the 1970s, while earlier studies by Kovba et al. [104] and Carnall et al. [105] suggested a rhombohedral and an orthorhombic symmetry, respectively. Gasperin who synthesized single crystals of Na2U2O7 by accident, while trying to make single crystals of UNb3O10 in excess Na2CO3, reported a rhombohedral structure (in space group R3m) with cell parameters a = 3.911(3) Å and c = 17.857(5) Å [88]. Thermodynamic investigations by Cordfunke et al. moreover suggested the existence of two reversible phase transitions for this phase above about 600 K (α→ β), and around 1348 K (β → γ), which they observed with a high temperature X-ray Guinier camera [102, 106]. But the corresponding X-ray diraction patterns and crystal structures were not reported. IJdo et al. solved the α and β structures in 2015 from powder neutron diraction data [91]. They reported a monoclinic symmetry, in space groups P 21/a and C2/m, for the α and β forms, respectively. They also suggested that the rhombohedral crystal structure, as deter- mined by Gasperin, could correspond to a high temperature modi cation, with delocalized U and O atoms [91].
There is very little data available on the Na6U7O24 (also written Na1.71U2O6.86) phase. 39
Table 3.1: Structural parameters of the sodium uranate phases.
Compound α-Na2UO4 β-Na2UO4 Na4UO5 Na2U2O7
Temperature (K) 298 298 298 298
Uranium Ox. State 6 6 6 6
Symmetry Orthorhombic Orthorhombic Tetragonal Rhombohedral
Z 2 4 2 3/2
Space group P bam(55) P bca (61) I4/m (87) R3m (166)
a (Å) 9.7623(3) 5.8079(3) 7.5172(1) 3.911(3)
b (Å) 5.7287(2) 5.9753(3) 7.5172(1) 3.911(3)
c(Å) 3.4956(1) 11.7179(6) 4.6325(2) 17.857(5)
Cell volume (Å3) 195.496(11) 406.650(34) 261.78(1) 236.55(1)
Reference [87] [87] [89] [88]
Compound α-Na2U2O7 β-Na2U2O7 Na6U7O24 NaUO3
Temperature (K) 293 773 298 298
Uranium Ox. State 6 6 6 5
Symmetry Monoclinic Monoclinic Rhombohedral Orthorhombic
Z 4 4 3/2 4 Space group P 21/a (14) C2/m (12) R3m (166) P bnm (62) a (Å) 12.7617(14) 12.933(1) 3.95(2) 5.7739(2) b (Å) 7.8384(10) 7.887(1) 3.95(2) 5.9051(2) c(Å) 6.8962(9) 6.9086(8) 17.82(2) 8.2784(2) β(○) 111.285(9) 110.816(10) 90 90 Cell volume (Å3) 642.78(10) 658.80(13) 240.79(10) 282.26(1) Reference [91] [91] [101] [90]
Compound m-Na3UO4 γ-Na3UO4 α-Na11U5O16 Na4UO4
Temperature (K) 298 298 298 298
Uranium Ox. State 5 5 4.2 4
Symmetry Cubic Cubic Cubic Cubic
Z 1 8 2 1 Space group F m3m (225) F d3m (227) P 4232(208) F m3m (225) a (Å) 4.77(2) 9.56(4) 9.543(2) 4.780(6) b (Å) 4.77(2) 9.56(4) 9.543(2) 4.780(6) c(Å) 4.77(2) 9.56(4) 9.543(2) 4.780(6) Cell volume (Å3) 108.53(10) 873.72(10) 869.07(1) 109.22(1) Reference [82] [84] [83] [86]
3.1. Introduction 41 Its X-ray diraction pattern was reported very similar to that of Na2U2O7. Cordfunke and Loopstra assigned it in 1971 to a triclinic C-face centred subcell, while the work of Toussaint and Avogadro [101] suggested a rhombohedral symmetry, in space group R3m, with lattice parameters a=3.95(2) Å and c=17.82(2) Å. No other studies have been reported since then. The existence of Na6U7O24 must hence be substantiated with complementary work, and the discrepancy regarding its crystal structure needs to be solved. Given the structural similarities between Na2U2O7 and Na6U7O24, one could expect a low temperature (α→ β) phase transition. The Na6U7O24 structure should have additional oxygen vacancies and 1/7 of the sodium sites vacant [107]. A third high temperature phase is not expected, however, as Cordfunke and Loopstra reported that Na6U7O24 was decomposing to Na2U2O7 and U3O8 in air above 1173 K [102].
Finally, the technological importance of the sodium urano-plutonate Na3(U1−α,Puα)O4 has led to a considerable interest for its structural [8285, 108] and thermodynamic [109 111] properties as mentioned in Chapter 1. The plutonium concentration in the (U,Pu)O2 fast reactor fuel being typically of the order of 20 wt%, Na3UO4 and Na3(U1−α,Puα)O4 are expected to be isostructural and have similar thermomechanical and thermodynamic properties (since the ionic radii of U5+ (0.76 Å) and Pu5+ (0.74 Å) in six-fold coordination are very close [112]) [812]. But the crystal structure of Na3UO4 has remained the subject of controversy until now [108].
Scholder and Gläser [82] rst reported in 1964 a disordered NaCl type of structure, obtained at low temperatures (T < 973 K), with cell parameter 4.77 Å (m phase). In 1970, Bartram and Fryxell [83] obtained, at temperatures ranging from 973 to 1273 K, a new phase named α, ordered this time with many additional re ections, and assigned it to cubic symmetry with a doubled cell parameter, 9.54 Å, and the chemical composition Na11U5O16. The latter composition was ruled out by Lorenzelli et al. [108] in 1985, however, as their X- ray diraction pattern showed many additional re ections not taken into account by Bartram and Fryxell, and that could not belong to the cubic structure. According to them, the correct composition is Na3UO4, though their attempts to index it in tetragonal and orthorhombic systems were not successful. In 1972, Marcon et al. [84] discovered the formation of a high temperature (>1273 K) partially ordered cubic phase of Na3UO4, named β, with the space group F d3m and cell parameter 9.56 Å. Lorenzelli et al. con rmed those results and established a reversible phase transition between the α and β forms around (1348 ± 25) K. Finally, Pillon revisited the question in 1989 [85] and 1993 [86] by means of X-ray diraction coupled to neutron diraction and microcalorimetry for quantitative analysis of the sodium formed during the reaction. Based on the latter quantitative analysis, she described the formation of a metastable intermediate tetravalent uranate Na4UO4 for temperatures below 848 K by direct reaction between UO2 and Na2O, followed by a decomposition reaction above 873 K with progressive loss of sodium towards the Na3+xUO4 composition. The structure of this metastable intermediate Na4UO4 phase was described as face centred cubic (f.c.c.),
very close to the structure of Scholder and Gläser [82], with a cell parameter 4.78 Å. The decomposition reaction to Na3UO4 was moreover shown to be irreversible. To the author's knowledge, no other studies have been reported since the work of Pillon et al. [86] and the discrepancies have remained.
In this rst result chapter, we have solved the structure of the α phase of the trisodium