Breaking the hard-sphere model with fluorite and antifluorite solid solutions

Using the hard-sphere model with the existing tabulated values of ionic radii to calculate the lattice parameters of minerals does not always match experimental data. An adaptation of this crystallographic model is proposed by considering the cations and anions as hard and soft close-packed spheres, respectively. We demonstrate the relevance of this “hybrid model” by combining Pauling’s first rule with experimental unit-cell parameters of fluorite and antifluorite-structured systems to revise the ionic radii of their constitutive species.

Ionic crystals are the alternate and periodic stacking of cations and anions forming a structural lattice by the balance between their attractive and repulsive forces. In the hard-sphere model, these ions are in contact and their radii are reported to depend on their oxidation state and coordination number (noted C.N.) 1 . Assessing these solids' lattice parameters from the ionic radii of the constitutive species is paramount to understanding their defect chemistry and/or variations in their composition. Pauling proposed five empirical rules that these crystals shall respect to be stable 2 . The first stipulates that the coordination number of the cation depends on the ionic radii ratio r cation /r anion and that the latter should range between specific values determined by geometry construction; otherwise, the structure is unstable, and the coordination changes. Shannon 3 highlighted that using the tabulated ionic radii to model such materials' lattice structures does not always match the experimental values and/or Pauling's rules. Some studies also propose revised ionic radii 4 that better agree with diffraction experiments; however, they do not coincide with the geometrical criteria.
Fluorites and antifluorites are critical ionic solids for a myriad of applications such as catalysts 5 , electroceramics 6 , or even nuclear fuels 7 , and thanks to their high symmetry, their geometry is simple to model (Fig. 1). Their lattice parameter a can be calculated using Eq. 1 from the body diagonal of the lattice (cube) and the ionic radii r of the constitutive species.
The cations and anions are in 8-and fourfold coordination in fluorites, respectively. The structure is stable if 0.732 ≤ r cation /r anion ≤ 1.000 8 . Since antifluorite is the fluorite's antistructure, these ions' positions are permuted and the crystallographic arrangement is stable if 0.225 ≤ r cation /r anion ≤ 0.414 8 . These structures were widely studied in the past century and experimental lattice parameters are numerous. At the light of these crystals, we break the hard-sphere model by demonstrating that anions are soft because their ionic radius not only depends on their charge and coordination number but also on the nature (thus size) of the closest neighbors. Table 1 shows Shannon's 1 ionic radii of the constitutive species of some selected important compounds for their use in various fields (fluorites: ZrO 2 , TbO 2 , HfO 2 , CeO 2 , ThO 2 , UO 2 , NpO 2 , PuO 2 , AmO 2 , CmO 2 , BkO 2 , CfO 2 , CaF 2 , SrF 2 , BaF 2 , EuF 2 , and PbF 2 ; antifluorites: Na 2 O, Li 2 O, and K 2 O). The associated r cation /r anion ratios, compared to the stability criteria of Pauling's first rule, and their experimental lattice parameters at room temperature obtained using diffraction methods are also listed.

Ideal fluorites & antifluorites
First of all, the difference in Pauling's electronegativity χ anion − χ cation in the selected compounds is larger than the minimum value of ⁓ 1.7 that defines the ionic solids 8,9 . Indeed, Δχ ranges between 2.06 (UO 2 ) and 3.09 www.nature.com/scientificreports/ (BaF 2 ). Discussing the ionic radii of the constitutive species is then legitimate and the high ionicity of the bonds allows approximating ions as spherical entities. The compounds highlighted in bold in Table 1 should not be stable if the r cation /r anion lower limit of Pauling's first rule is respected or if Shannon's ionic radii are correct. To fulfill the geometric stability criteria and to match the experimental unit-cell measurements of these ionic crystals, we consider the anions as soft spheres with an effective radius varying as a function of the cation's nature by deriving Eq. (1) in Eq. (2).  Table 1. Calculated r cation /r anion ratios from Shannon's ionic radii 1 of selected fluorite and antifluoritestructured compounds and their experimental lattice parameters at room temperature determined using X-ray diffraction. *BaF 2 is a peculiar example as the cation is larger than the anion (r cation /r anion > 1). From a geometrical stability point of view, it shall be considered as an antifluorite as the smaller ion is in fourfold coordination instead of 8. The ratios highlighted in italic and bold correspond to the structures that respect and violate Pauling's first rule, respectively. Pauling's electronegativity χ 9 values and χ anion − χ cation are also listed. www.nature.com/scientificreports/ In the peculiar case of the fluorites highlighted in bold in Table 1, the r cation /r anion ratio is smaller than the lower stability limit, so cations and anions are not in contact in this configuration. The unit-cell is a close-packed arrangement of anions and can be calculated using Eq. 3.
If a fixed anionic radius is considered, these compounds' lattice parameters will also be, geometrically, independent of the nature of the cations that partly fill the interstices of the structure (Fig. 1). However, this is not verified experimentally, as evidenced by the variations in the lattice parameters observed when the cation is changed ( Table 1). The cations and anions shall be, at least, in contact one another 2 . Thus, from the accurate experimental lattice parameters, applying Eq. (3) to UO 2 and PuO 2 gives two oxygen radii of 1.368 and 1.349 Å, respectively. Implementing these values in Eq. (2) allows determining the revised cation radii giving 1.001 and 0.987 Å for U(IV) and Pu(IV) in 8-coordination, respectively. Similarly, the constitutive species' ionic radii of the bold fluorite compounds in Table 1 were re-evaluated ( Table 3 in supplementary materials). UO 2 and PuO 2 are known to form a solid solution of U 1−y Pu y O 2 , i.e., U and Pu can be substituted in their mutual lattice. When the two oxygen ionic radii are individually used to plot the theoretical Vegard's law between UO 2 and PuO 2 , no value matches the experimental variations in the fluorite structure's lattice parameter as a function of the Pu content, y (Fig. 2a). As the commuted species, U and Pu do not have the same ionic radii, the crystal's geometry is locally deformed. Since the ions are considered contacted spheres, the cation-anion distance will change when incorporating the doping species in the host lattice. To do so, and by considering the anions as a close-pack arrangement, the r anion will vary proportionally to the incorporation of the doping cation within the lattice. Figure 2b plots the variations in the oxygen ion size as a function of the plutonium concentration in the U 1−y Pu y O 2 solid solution from the linear regression between UO 2 and PuO 2 . This model allows reproducing of the experimental lattice parameters of U 1−y Pu y O 2 and matches the geometrical criteria established by Pauling.
As antifluorites are the antistructure of fluorites, the same method can be used to estimate their solid solutions' lattice parameters from the pure constitutive poles.
If the coordination number and the oxidation state of the considered cations do not change, this method could be used for any ionic crystal and predict the lattice parameters of solid solutions not yet investigated experimentally, as long as the difference in cations' electronegativity, Δχ M , is small. Indeed, a large Δχ M results in a disordered structure due to the resulting different Cation-Anion bond distances. Ultimately, the gradual incorporation of a dopant element within the host lattice changes the local coordination number of one of the cations, even if they have identical oxidation sates and comparable ionic radii 27 .

Incorporating aliovalent cations
Doping is widely used for tailoring a given material's properties (optical, electrical, redox). Aliovalent atoms are frequently used as dopants to boost the target properties. Also, some cations can have various oxidation states within the same structure, resulting in a deviation from stoichiometry. Due to their exceptional aptitude to form solid solutions, fluorites and antifluorites are often doped, even with aliovalent cations, and can likewise be subjected to dramatic variations in stoichiometry 7,16 and/or complex charge compensation processes 28 . Such www.nature.com/scientificreports/ intrusive atoms generate lattice distortions and/or crystal defects, and the ratio of the ionic radii of the two permuted atoms will be considered to evaluate the effect of such a dopant on the host structure's lattice parameters. However, using our method in such materials might be problematic because one of the pure poles might crystallize in a different structure than the host lattice. For instance, in Nd-doped UO 2 fluorite, neodymium is trivalent, and its oxide form is Nd 2 O 3 (either cubic or haxagonal 29 ); thus, a direct application of our method cannot estimate the variations in the oxygen radius with the Nd concentration in U 1−y Nd y O 2 . Therefore, we propose an alternate method that compares the fluorite/nonfluorite pseudo-binary system to a known fluorite/fluorite couple. Hence, our UO 2 /Nd 2 O 3 example can be paralleled to the well-known UO 2 /PuO 2 system by comparing the sizes of Pu(IV) and Nd(III) ions in eightfold coordination. Also, doping UO 2 with trivalent neodymium should be balanced by a partial oxidation of uranium to its pentavalent state in the same proportions as the Nd incorporation 30 . Thus, the size of U(V) will be considered. Table 2 presents the ionic radii of the constitutive cations (C.N. = 8) and the associated r cation /r Pu(IV) ratios. Figure 3a plots the variations in the experimental lattice parameters of U 1−y M y O 2 as a function of the dopant content y with M = Pu 16,[33][34][35][36] or Nd 30,31,[37][38][39] , and (b) represents the calculated theoretical lattice parameters of U 1−y Nd y O 2 from the ionic radii ratios of Table 2.
The exquisite agreement between the experimental and calculated lattice parameters of U 1−y Nd y O 2 represented in Fig. 3b confirms that: (1) the theoretical crystallography calculations verify that the Nd(III)/U(V) charge compensation model in U 1−y Nd y O 2 is correct, and (2) our geometrical hybrid model based on the parallelism between the considered fluorite/nonfluorite system to a known fluorite/fluorite one is successful. However, one may keep in mind that the disorder generated by the incorporation of a dopant cation (aliovalent or not) within the lattice may affect the unit-cell parameters, inducing deviations from the pseudo-Vegard's law 40 .
Similarly, the investigation of higher-order systems is possible. For example, stoichiometric U 1−y M y O 2 with M = Pu + Am is a complex system where americium can take a pure tetravalent oxidation state, a mixed Am(III)/ Am(IV) valence, or be purely trivalent depending on its concentration and distribution homogeneity 41 . Likewise, Am(III) in uranium-plutonium-americium mixed oxides is balanced by U(V) 28 , but a clear trend regarding the  www.nature.com/scientificreports/ Am(III)/Am ratio remains to be determined. In pure americium dioxide, the r cation /r anion is smaller than Pauling's geometrical limit (Table 1), and the anions should touch and form a close-pack arrangement. Since the lattice parameter of AmO 2 at room temperature is 5.3755(5) Å 16 , the ionic radius of O(-II) in this compound is 1.344 Å, and gives, using Eq. (2), the revised cation radius of r Am(IV) = 0.984 Å (for C.N. = 8). The ionic radius of Am(III) in eightfold coordination is taken from Cross 42 and equal to 1.108 Å. From these ionic radii and by using the same method as above, the variations in the lattice parameters of U 1−y {Pu 1−α [Am(IV) 1−β Am(III) β ] α } y O 2 as a function of plutonium, americium, and Am(III)/Am contents can be calculated using Eq. (4).
with the oxygen ionic radius r O(−II)U−Pu−Am y, α, β calculated in the same manner as for the Nd-doped UO 2 .

Deviation from stoichiometry in fluorite-structured oxides
In addition to their ability to form solid solutions, the fluorite structure (oxides) can accommodate large deviations from stoichiometry 7,16,43 . We enlarged our hybrid model to nonstoichiometric materials. In oxygen-hypostoichiometric fluorite-structured dioxides (Oxygen/Metal < 2), Kim 44 and Chatzichristodoulou 45 have envisaged a flexible oxygen vacancy size but considered a fixed ionic oxygen radius. Since hypostoichiometry corresponds to removing anions from the crystal, the associated reduced cations' coordination number should, theoretically, be modified likewise (Fig. 1). However, due to the Born-Haber energy, the lattice does not collapse, therefore, we considered the oxygen vacancies as spherical entities instead of empty voids. The coordination number of the cations remains unchanged and equal to 8.
For discussion, we examined the fluorite-structured hypostoichiometric uranium-plutonium mixed oxides U 1−y Pu y O 2−x for which an empirical relation between the experimental lattice parameters (in Å) and deviations from stoichiometry at room temperature 31,46 exists. It is updated in Eq. (5) from the recent measurements of the accurate lattice parameters of UO 2 15 and PuO 2 16 .
In U 1−y Pu y O 2−x , Pu(IV) can be partially reduced to Pu(III) and is solely responsible for the hypostoichiometry in the mixed oxide below ⁓ 1700 K. Due to the solid's electroneutrality, the deviation from stoichiometry x is directly correlated to the valence of the constitutive cations. If we simplify the system by focusing on the PuO 2−x dioxide (so U 1−y Pu y O 2−x with y = 1) and considering that the oxygen vacancies are doubly charged and balanced by two Pu(III), PuO 2−x can be defined as Pu(IV) 1−2x Pu(III) 2x O 2−x . Since tetra-and trivalent Pu have different ionic radii in the eightfold coordination, PuO 2−x will be treated as a solid solution of the hypothetical Pu(IV)O 2 -Pu(III) O 2 system (even if the second end member has no physical meaning). Before estimating the oxygen vacancy's effective size in PuO 2−x , the oxygen ionic radius r O(-II) must be determined. Therefore, the ionic radius of Pu(III) in the eightfold coordination, taken from Cross 42  These results confirm that the oxygen, vacancy's effective size is not constant, likewise the ionic radius of oxygen, and it depends on the dopant's concentration, here Pu(III), and rapidly decreases and stabilizes with the increasing deviation from stoichiometry.
We believe this method can be used for more complex systems, including solid solutions and/or compounds with aliovalent cations, such as plutonium in PuO 2−x .

Conclusions
This innovative method cleaves with the globally adopted model of ionic radii only depending on the constitutive species' oxidation state and coordination number. Using simple structures, such as fluorites and antifluorites, we break the hard-sphere model by demonstrating that the anions are soft because their radius varies with the nature (size) of the surrounding cations. This hybrid model does not violate Pauling's first rule and can predict the lattice parameters of new fluorite and antifluorite-structured solid solutions if the pure poles are known, as long as the host and dopant metal atoms have similar electronegativities. Finally, this geometrical method's transposition is possible to investigate the deviations from stoichiometry. This result confirms that, in hypostoichiometry, the oxygen vacancy's effective size will not be considered constant and depends on the doping species' nature and concentration. We also reasonably envisage that this method could be used for any ionic crystal.  x was only set up to 0.02 because PuO 1.98 is the more reduced composition plutonium dioxide can reach without changing its structure at room temperature, i.e., without forming a second phase.