Correct prediction of structural parameters by a simulation method is a key factor for atomistic modeling-based characterization of materials. The force fields used in classical molecular dynamics simulations are often parameterized to reproduce structural parameters of investigated materials, and this has been also the case in the research on pyrochlores (e.g. Minervini et al., 2000). In similar way, the ab initio and, in principle, DFT simulations, are also expected to provide a good description of the structures of solids. However, different applied approximations (e.g., exchange-correlation functionals) result in different quality of the computed structural data (Perdew et al., 2008; Blanca-Romero et al., 2014). In a series of papers on orthophosphates we demonstrated that a very good match to the structural parameters, especially the Ln-O bond lengths, can be obtained with the PBEsol exchange-correlation functional (Perdew et al., 2008) and the DFT + U method with the Hubbard U parameter derived from first principles (Blanca-Romero et al., 2014; Beridze et al., 2016; Kowalski et al., 2021). Finkeldei et al. (2017) have shown that PBEsol functional can predict accurate value of the lattice parameter of the Nd₂Zr₂O₇ pyrochlore. We note that those calculations were performed with the f in the core approach (see Section 2). With the PBEsol exchange-correlation functional and the outlined parameter-free DFT + U method Blanca-Romero et al. (2014) got excellent results for the Ln−O bond-lengths in and volumes of LnPO₄ compounds. Here we test if such a method could predict good lattice parameters and Ln−O bond-lengths for series of zirconate-pyrochlores. In Table 1 we provide a set of the Hubbard U parameters computed for different lanthanide cations. The values are consistent with the previous simulations of Li et al. (2015) for pyrochlores and the aforementioned results of Blanca-Romero et al. (2014) for lanthanide phosphates. There is a clear trend that reflects strength of the electronic correlations for different Ln cations. The Hubbard U parameter increases along lanthanide series, reaching maximum for Eu and then becomes smaller for Gd. This is an effect of completely filled half f-shell of Gd³⁺.

An important test of a computational method is its ability to reproduce the measured lattice parameters of computed crystalline solids. Although some studies report excellent match of the computed lattice parameters of pyrochlores to the measured values (Feng et al., 2011), our previous studies have shown that the structural parameters of lanthanide-orthophosphates are very sensitive to the applied computational method, especially to the exchange-correlation functional (Blanca-Romero et al., 2014). A correct computation of strongly correlated 4f electrons also plays an important role in those cases. The lattice parameters of considered stoichiometric pyrochlores have been measured at ambient conditions by different studies (Sasaki et al., 2004; Harvey et al., 2005; Mandal et al., 2007; Nästren et al., 2009; Hagiwara et al., 2013; Koohpayeh et al., 2014; Vaisakhan Thampi et al., 2017). These are collected in Figure 2 and compared to the computed data. With our computational setup, applying the PBEsol exchange-correlations functional, we got excellent match to the measured lattice parameters of all the considered stoichiometric zirconate pyrochlores, with a difference within ∼ 0.04 A ̊ . We note that thermal expansion contributes to this difference at the level of ∼ 0.01 A ̊ (see Section 3.1.3). Such a good result is very important when, for instance, analyzing the structural change upon doping pyrochlores with actinide elements (Section 3.1.3). On the other hand, it is clear that the PBE exchange-correlation functional overestimates the lattice parameter by ∼ 0.12 A ̊ .

The lattice parameter of non-stoichiometric fluorite and pyrochlore compounds can be predicted with an ion-close-packing approach by taking into account charges and coordination numbers of A and B cations and the presence of oxygen vacancies (Ohmichi et al., 1981; Hong and Virkar, 1995; Marrocchelli et al., 2012, 2013; Bukaemskiy et al., 2021). The fluorite lattice parameter, a, can be expressed as a function of the average radii of cations and anions as follows a = 4 3 R c + R a , (1) where R _c and R _a are additive sums of contributions from all types of cations and anions. The pyrochlore lattice parameter is twice the fluorite lattice parameter. It has been recognized that a vacancy can be viewed as an anion species with a certain effective radius. Models, which apply this approach to fluorite compounds are divided into two groups. In the first group, the vacancy radius is found to be significantly smaller than the radius of an oxygen anion (Hong and Virkar, 1995; Marrocchelli et al., 2012; Marrocchelli et al., 2013), in the second group the vacancy is significantly larger than the oxygen anion (Ohmichi et al., 1981; Bukaemskiy et al., 2021). As discussed recently by Bukaemskiy et al. (2021), the distinction is due to taking or not taking into an account a change in the average cation coordination due to the insertion of vacancies. If the average cation coordination is counted properly as 8 − 2x, then the vacancy radius is predicted to be about 11% larger than the ionic radius of O²⁻. The later approach is shown to be consistent with the results of ab initio simulations (Stapper et al., 1999; Bukaemskiy et al., 2021). The loss of the negative charge of an oxygen anion (due to a vacancy formation) pushes the four cations that form tetrahedron encapsulating the 8a vacancy (Figure 1) away by ∼ 0.18 Å, indicating the vacancy size to be larger than O²⁻. On the other hand, the six nearest oxygen atoms move towards the vacancy (and to the four cations nearest to the vacancy) by ∼ 0.24 A ̊ . In the first group of models, both the extension and the contraction effects are mapped onto the vacancy size thus making its radius smaller than that of O²⁻. In the second group of models, the latter contraction effect is taken into account via a change in the cation radii. The coordination numbers of the four cations are decreased due to the vacancy formation, consequently their cation radii decrease. The average cation coordination number in the stoichiometric pyrochlore is thus 7 (8 – 2x for x = 0.5), while it is eight in the ideal fluorite (x = 0). Models that take into account the change in cation coordination allow a more accurate description of the lattice parameter variation in fluorite and pyrochlore compounds, because they are sensitive not only to changes in the average composition, but also to short- and long-range order effects. For example, due to the long-range order in pyrochlore, the coordination numbers of A and B cations appear to be significantly different from the average value of 7. The B cation is 6-fold coordinated, while the A cation is 8-fold coordinated. As shown by Bukaemskiy et al. (2021), the tendency of B cations to lower their coordination number, i.e. to attract vacancies, can be deduced to exist at non-stoichiometric compositions as well. For example, the lattice parameter variation in A _x B _1−x O_2−0.5x systems (A = Ln, Y; B = Zr) within the interval of 0 < x < 1/3 can be explained under the assumption that all vacancies are fully surrounded by B cations and all A cations keep their coordination equal to 8. It is further shown that this strategy breaks down at x > 1/3, when the amount of Zr is insufficient for building up the surrounding of isolated vacancies. Within the interval of 1/3 < x < 0.5 two different models of SRO develop, one of which is consistent with the type of order in pyrochlore. The tendency of a (large) vacancy to reside close to a B cation is valid only in systems in which a B cation is significantly smaller than an A cation. When the radii of A and B cations are approximately equal, the electrostatic tendency of a vacancy association to an A (III) cation becomes more important. This is confirmed by the ab initio simulations (Bogicevic et al., 2001; Bogicevic and Wolverton, 2003; Solomon et al., 2016).

Finkeldei et al. (2017) investigated the series of Nd_xZr_1-xO_2-x/2 compositions by applying XRD and calorimetric measurements, and ab initio modeling methods. The XRD data show gradual, linear decrease in the lattice parameter upon lowering of Nd content with x, showing slightly different slopes within fluorite (0.23 < x < 0.30) and pyrochlore (0.33 < x < 0.53) domains. The observed trend for pyrochlore compounds has been well reproduced by the computed data, applying a structural model in which with gradual decreasing of Nd content, the oxygen vacancies are randomly filled with oxygen atoms.

When stoichiometric pyrochlore is doped with actinide elements, the main questions are the incorporation site (A or B cation) and the oxidation state of the dopant. The experimental data on Nd₂Zr₂O₇ pyrochlore of Nästren et al. (2009) and Finkeldei et al. (2020) indicate doping at the A cation site (Nd), even when the oxidation state of An cation is larger than 3+ . In order to shed light on this phenomenon we performed calculations of the lattice parameter of a series of Nd₂Zr₂O₇ pyrochlores doped with different actinide elements (An = U, Np, Pu, Am, Cm). In the first step we computed the Hubbard U parameters for actinides incorporated into the Nd-pyrochlore at different cation sites and in different oxidation states. The values are listed in Table 2. The computed values are consistent with the results of our previous studies (e.g., Beridze and Kowalski (2014); Beridze et al. (2016); Kvashnina et al. (2018); Sun et al. (2021)). There is an increase in the Hubbard U parameter with the oxidation state of An cation, and a general increase along actinide series, which is similar to the trend obtained for the Ln series (Table 1). This has an impact on the lattice parameters of An-doped pyrochlores.

Figure 3 shows the computed and measured (Nästren et al., 2009) lattice parameters of Nd₂Zr₂O₇ pyrochlore doped with different actinides. In order to be consistent with the measurements of Nästren et al. (2009), all the computed values were re-scaled to actinide stoichiometries of 0.2 per A ₂ B ₂O₇ formula unit and were corrected for thermal expansion using the linear thermal expansion coefficient (TEC) and assuming ambient temperature of 298 K. There is some disagreement in the literature about the low temperature (from 0 K up to 300 K) TEC of neodymium zirconate pyrochlores (Kutty et al., 1994; Shimamura et al., 2007; Liu et al., 2010; Feng et al., 2012; Guo et al., 2015) (range from 5 ⋅ 10^–6 K⁻¹ to 12 ⋅ 10^–6 K⁻¹). Here we applied the representative value of 9.11 10^–5 K⁻¹ determined by Kutty et al. (1994). We note, however, that due to the uncertainty in the TEC, the lattice parameters values computed here could have an additional error of ∼ 0.01 Å. Nevertheless, with the selected TEC we perfectly reproduce the measured lattice parameter of stoichiometric Nd₂Zr₂O₇ pyrochlore, which is essential when predicting the change in the lattice parameter upon incorporation of actinide elements.

The measurements of Nästren et al. (2009) show a clear linear-like relationship between the lattice parameter of An-doped Nd₂Zr₂O₇ and the ionic radius of the An cation. Figure 3 shows that when the f electrons are not computed explicitly, this trend is captured only on the qualitative level. In contrast, the lattice parameters computed with the parameter-free DFT + U method used here, with the Hubbard correction applied to An and Ln elements, successfully reproduce the linear relationship identified by Nästren et al. (2009). This shows the importance of electronic correlations on reliable and accurate computation of lattice parameters of f elements containing oxides. The calculations show that tetravalent and pentavalent actinides, when doped on the Nd³⁺ sites, cause a significant lattice contraction, while tetravalent actinides when doped on the Zr⁴⁺ site lead to significant lattice expansion. The later effect occurs because 6-fold coordinated Zr⁴⁺ has the ionic radius of only 0.72 Å (Shannon, 1976), which is significantly smaller than that of the considered An cations. As the measurements of Nästren et al. (2009) show a lattice contraction for all tetra and pentavalent actinides, the excellent match between the values computed here and the experimental data supports their interpretation that the actinides cations are incorporated on the Nd-site.

On the computational side, the DFT + U method improves accuracy of the predicted lattice parameters, reducing the average error from 0.017 Å (obtained with the simple DFT-PBEsol approach (f in the core) to just 0.005 Å (i.e. less than 0.1%), which is an unprecedented accuracy having in mind that 4f and 5f elements are computationally challenging. Pu³⁺ is also correctly predicted to expand the lattice. Furthermore, the prediction of the lattice parameters of pristine material (Nd³⁺) and that of Pu³⁺ doped systems have relative errors of only 0.001 and 0.09%, respectively. Even considering the uncertainty in the TEC, these are excellent results showing that the parameter free DFT + U method is a highly reliable tool for predicting lattice parameters in the actinide-pyrochlore system.

An interesting observation is that doping with Cm⁴⁺ produces a significantly greater lattice parameter than with Am⁴⁺, breaking the linear relationship followed by the other An cations’ cases. This is not expected as Cm⁴⁺ has smaller ionic radius than Am⁴⁺. We note, however, that such a subtle differences have been observed due to so-called tetrad effect associated with the number of electrons in f shells (e.g., Wilden et al. (2019)). On the computational side, the reason for this is a significantly larger Hubbard U parameter value derived for Cm⁴⁺ than that for Am⁴⁺ or Cm³⁺ (Table 2). This resembles the case of Eu³⁺ and Gd³⁺ (Table 1) and results from the strongest electronic correlation effects for cations with nearly complete f half-shell (i.e., with 6 f electrons). Unfortunately, we do not have experimental data for Cm doped Nd₂Zr₂O₇ in the data set of Nästren et al. (2009) to confirm our claim. Confirmation of such effect would be a direct demonstration of impact of electronic correlation effects on the structural arrangements in f elements oxides.

Empirical models can be applied to predict the change in lattice parameters of doped materials. The ion-close-packing model of Bukaemskiy et al. (2021) provides information on the structural effects occurring in a doped compound purely due to charges in coordination numbers of the constituent cations. Figure 3 shows the variation of lattice parameters of actinide-bearing Nd-pyrochlore computed with the ion-packing model, utilizing the Shannon’s cation radii (Shannon, 1976). The model provides values that are in good agreement with the experimental lattice parameter data and the ab initio calculations discussed above (although it can not capture the discussed, subtle effects resulting from the electronic correlations). This is yet another confirmation of the A cation site doping scenario irrespectively of the valence state of An cations. On the other hand, the empirical model suggests that the linear relationship observed by Nästren et al. (2009) between the lattice parameter and the radius of An cation is split into three linear trends corresponding to +3, +4 and +5 cations.

The formation enthalpies from oxides for series of stoichiometric pyrochlore-type compounds were measured by Helean et al. (2004) (Ln₂Ti₂O₇), Lian et al. (2006a) (Ln₂Sn₂O₇), Ushakov et al. (2007) (Ln₂Hf₂O₇) and Radha et al. (2009); Saradhi et al. (2012) (Ln₂Zr₂O₇). Kowalski (2020) derived these values with the DFT-based atomistic modeling methods. He was able to reproduce the experimentally seen trends in the behavior of formation enthalpies along the lanthanide series with the derived values within 20 kJ/mol. Accurate computation of formation enthalpies is crucial for predicting other important parameters, such as for instance, the order-disorder transition temperature (Jiang et al., 2009; Li et al., 2015) and the extend of ordering in disordered pyrochlore compounds.

Atomistic modeling studies have been applied to predict the order-disorder transition temperature (T _O−D ) of series of stoichiometric pyrochlore compounds. This parameters is usually computed assuming simple relationship between the enthalpy ΔH and entropy ΔS of disordering process: T O − D = Δ H Δ S , (2) and an estimate of the disordering entropy assuming a complete disordering of the disordered phase (e.g., Jiang et al. (2009)). Earlier atomistic modeling studies derived this parameter by applying simple force-fields to describe interatomic interactions (Minervini et al., 2000; Sickafus et al., 2000). Jiang et al. (2009) applied DFT to derive T _O−D and obtained values that are smaller by ∼300 K from the measured values, but the experimentally observed trend in decrease of transition temperature along lanthanide series has been well captured. The main reason for this offset is the application of the standard DFT method with the f electrons included into the pseudopotential core and not computed explicitly. Li et al. (2015) have shown that the prediction of T _O−D is significantly improved when f electrons are computed explicitly and the electronic correlations computed with the DFT + U method with the Hubbard U parameter derived from first principles using the linear response method (Cococcioni and de Gironcoli, 2005). They got a nearly perfect agreement with the measured T _O−D for Sm₂Zr₂O₇, Gd₂Zr₂O₇, Gd₂Hf₂O₇ and Tb₂Hf₂O₇ compounds.

The aforementioned computed results may be seen surprising in view of the more recent finding of significant SRO in this class of compounds (Shamblin et al., 2016). We notice however, that a SRO is accompanied by a simultaneous decrease in the disordering enthalpy and entropy, as demonstrated by Kowalski (2020). This results in preservation of constancy of the ratio given in Eq. 2. Furthermore, as shown by Finkeldei et al. (2017) for Nd₂Zr₂O₇, the correct prediction of T _O−D may require application of more complex model of disordered phase. In that study a simple two-state statistical model has been applied. Results of applying such a model to selected pyrochlore compounds is given in Table 3. We note that the prediction could be improved with: 1) a better computational setup (explicit computation of f electrons, Li et al. (2015)) and 2) with a better structural model of disordered pyrochlore, and disordering entropy in such a system. The later would require more advanced thermodynamic modeling approach.

The extend of SRO in stoichiometric defect fluorite compositions has been demonstrated experimentally with calorimetric measurements (Ushakov et al., 2007; Saradhi et al., 2012; Finkeldei et al., 2017) and neutron scattering data (Shamblin et al., 2016). The later data are best fitted with a weberite-type structural representation, in which SRO is emulated via a long-range ordered phase that includes both ordered and disordered sublattices. This finding has been supported by atomistic simulations of Kowalski (2020) who has shown that formation enthalpies computed with the weberite-type structural model are in good agreement with the aforementioned calorimetric measurements and that fully disordered defect fluorite structure is thermodynamically unfeasible. The fact that weberite-type structural model fits nicely the data, however, is not a direct proof that this model gives the best possible characterization of cation and anion distribution in the disordered pyrochlore. Theoretical considerations lead to other possible structural arrangements (Bukaemskiy et al., 2021; Vinograd and Bukaemskiy, 2021). The weberite model may not be the best solution for the configurational entropy of the disordered phase, as the nominal entropy effect of a pyrochlore/weberite transformation (11.52 J/K/mol) seems too small to fit the calorimetric data for Eu₂Zr₂O₇ (18 J/K/mol, Saradhi et al. (2012); Finkeldei et al. (2017)). Nevertheless, the data for Gd₂Hf₂O₇ (12 J/K/mol, Ushakov et al. (2007)) seem to be consistent with the weberite scenario. The recent simulation study of Vinograd and Bukaemskiy (2021) provided a thermodynamic model for the disordered fluorite, which counts the configurational entropy contributions only from A and B cations occurring in the same coordination. The model assumes that the disordered fluorite is characterized by an ordered distribution of vacancies and a random distribution of A and B cations that all occur in the 7-fold coordination. The resulting nominal entropy of 23.04 J/K/mol is larger than the aforementioned experimental data, but such a model can not be easily rejected considering the likely presence of a degree of disorder in measured pyrochlore samples, which would lead to underestimation of disordering entropy by the calorimetry studies. Nevertheless, the model of Vinograd and Bukaemskiy (2021) provides a semi-quantitative description of the solution calorimetry data at non-stoichiometric compositions of A _x B _1−x O_2−0.5x systems with A = Sm, Gd, Dy and Y. It predicted the stabilization of various types of short and long-range order, depending on the temperature and the composition.

Finkeldei et al. (2017) studied the composition-induced disordering in Nd_xZr_1-xO_2-x/2 pyrochlore by applying XRD, calorimetric and atomistic modeling methods. On experimental side they were able to detect the formation of a disordered fluorite phase at x = 0.31 (corresponding to Zr/Nd ratio of 2). This has been clearly indicated by a small change in the lattice parameter vs. x slope, within a narrow transition region in which the two phases with slightly different lattice parameters coexist, and by a significant increase in the measured formation enthalpy of ∼30 kJ/mol, which corresponds to the disordering entropy of ∼16 J/K/mol. The later value is significantly smaller than the value expected for the fully disordered defect fluorite phase of ∼24 J/K/mol and clearly indicates the presence of a significant SRO. Finkeldei et al. (2017) introduced a disordering model in which, upon decrease of Nd content, vacant sites are gradually and randomly occupied by oxygen atoms (coming from excess of ZrO₂). They were able to reproduce well the change in the lattice parameter and formation enthalpy of pyrochlore phase, but significantly overestimated the formation enthalpy of defect fluorite compounds series. Following experimental finding of weberite-type SRO by Shamblin et al. (2016), Finkeldei et al. (2017) modelled the disordered phase as a statistical mixture of weberite and ideally disordered defect fluorite. With this approach, the measured data could be reproduced reasonably well, which shows yet another evidence for significant SRO in this class of compounds and demonstrates that weberite is a reasonable model for this phenomenon. On the other hand, these simulations have shown that weberite model, providing a good structural description of local ordering in defect fluorite, may not be a perfect model for predicting configurational entropies, implying the need for an additional (more disordered) structural component.