Thermodynamic assessment of the Ni–Te system
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Abstract
A thermodynamic assessment of the Ni–Te system has been performed using the Calphad method, based on experimental data available in the literature. The proposed description has been developed for use in the modeling of fission-product-induced internal corrosion of stainless steel cladding in Generation IV nuclear reactors. DFT calculations were performed to obtain 0 K properties of solid phases to assist the thermodynamic optimization. The ionic liquid two-sublattice model was used, and most solution phases were modeled using interstitial metal sub-lattices. With a strict number of parameters, the resulting description satisfactorily reproduces all thermodynamic properties and high-temperature phase transitions. The metastable miscibility gap in the Ni-rich liquid that is experimentally suggested is not present in the final description. The \(\delta \) phase exhibits a metastable order-disorder transition between the CdI\(_{2}\) and NiAs types of interstitial nickel distribution. The CdI\(_{2}\) prototype is the stable space group at room temperature. Low-temperature ordering phase transitions have been disregarded in this description, since they are not of interest to the application of corrosion in nuclear reactors.
Introduction
Internal corrosion of stainless steel fuel pins for Generation IV nuclear reactors, induced by the fission products Cs and Te [1, 2, 3, 4, 5, 6], might limit the lifetime of the type of reactor, and the corrosion must be modeled in order to predict its potential impact on the fuel pin integrity. Therefore, a thermodynamic database of transition-metal tellurides is underdevelopment to be incorporated into the thermodynamics of advanced fuels-international database (TAF-ID) [7]. Descriptions of the Fe–Te and Fe–Ni systems are available in the literature [8, 9], and in order to model the Fe–Ni–Te system, Ni–Te has to be assessed and that is the topic of the present paper. DFT calculations were performed in order to get an estimate of the formation enthalpy and lattice stabilities of certain Ni–Te compounds.
State of the art on the Ni–Te system
Phase diagrams and crystallography
The phase relations in the Ni–Te system have been studied by a few authors [10, 11, 12, 13, 14, 15]. Lee and Nash [16] assessed the data to construct the most recent phase diagram as seen in Fig. 1. Crystallographic data have been well studied for the \(\beta 2\), \(\gamma 1\) and \(\delta \) phases (Table 1) [12, 17, 18, 19].
The intermediate phases \(\delta \), \(\beta 2\) and \(\gamma 1\) consist of almost close-packed lattices of Te-atoms with Ni partially occupying interstitial sites. The \(\beta 2\) phase has at high temperature a Cu\(_{2}\)Sb-type structure with additional interstitial Ni partially occupying the Ni(2) octahedral site [12, 17]. The phase has two different ordered structures at the low-temperature solubility limits: The nickel-rich structure is monoclinic, below \({140}^\circ \hbox {C}\) [12], and the tellurium-rich one is orthorhombic, below 190–310 \(^\circ \hbox {C}\) [20]. The close relation of these structures to the Cu\(_{2}\)Sb-type allotrope is well described by Gulay et al. [17], who in a later paper determined the structure of the \(\gamma 1\) phase [19]. Several authors have verified that the \(\delta \) phase experiences an order-disorder transition at about 54.8 at.% Te [18] and a maximum in the c-axis, with the Ni-rich side being of the NiAs-type disordered structure and the Te-rich side of CdI\(_{2}\)-type ordering of defects into every other interstitial layer [12, 13, 18, 21, 22, 23]. Barstad et al. [13] reported that there is no solubility of Te in FCC–Ni based on lattice parameter measurements, and Abakarov et al. [24] published a solubility of 0.022 at.% Ni in Te at room temperature based on the same method.
The \(\beta 1\) and \(\gamma 2\) phases decompose on quenching, hence the lack of data. The \(\beta 1\) phase has an ordered superstructure of doubled axes, \(\beta 1'\), on the Ni-rich side between 731 and \({793}^\circ \hbox {C}\) [16]. Stevels [20] presented in a thesis high-T XRD data on the \(\beta 1\) phase and its ordered allotrope \(\beta 1'\). With calculations, they reproduced powder patterns of the \(F\bar{4}3m\) space group with varying metal-site occupancies and got a good description with 3.8 Ni-atoms in 4(c), 1.2 in 4(b) and 1 atom in 16(e). They compared this with the \(\beta \)-Cu\(_{2}\)Se phase of very similar lattice parameter of a = 5.69 Å, but that phase has later been re-characterized as \(Fm\bar{3}m\) [25]. The essential difference is that in \(Fm\bar{3}m\) the number of clusters of 4(c) sites each surrounded by four 16(e) sites is doubled by vertical mirroring.
Crystallographic data on phases of the Ni–Te system available in the literature
Phase | Pearson symbol | Space group | Lattice parameters [Å] | |||||
---|---|---|---|---|---|---|---|---|
a | b | c | \(\beta {^\circ }\) | Prototype | References | |||
\(\gamma -Ni\) | cF4 | \(Fm \bar{3} m\) | Cu | |||||
\(\beta 1\) | cF28* | \(Fm\bar{3}m/F\bar{4} 3m\)* | 5.71 | – | – | – | \({\rm Cu}_{2}{\rm Se}\)* | [20] |
\(\beta 1'\) | N/A | N/A | 11.44 | – | – | – | N/A | [20] |
\(\beta 2\) (ht) | tP6 | P4 / nmm : 2 | 3.99 | – | 6.09 | – | \({\rm Cu}_{2}{\rm Sb}\) | |
\(\beta 2\) (hn) | mP8 | \(P2_{1}/m\) | 7.54 | 3.79 | 6.09 | 91.16 | N/A | [17] |
\(\beta 2\) (ln) | oS10 | Pma2 | 7.54 | 3.79 | 6.06 | – | N/A | [17] |
\(\delta \) | hP3 | \(P\bar{3}/m1\) | 3.97 | – | 5.36 | – | \({\rm CdI}_{2}\) | [18] |
hP4 | \(P6_{3}/mmc\) | 3.86 | – | 5.26 | – | NiAs | [18] | |
\(\gamma 1\) | oP19 | Pnma | 12.38 | 3.92 | 6.88 | – | \({\rm Ni}_{1.10}{\rm Se}_{0.16}{\rm Te}_{0.74}\) | [19] |
\(\gamma 2\) | N/A | N/A | N/A | |||||
Te | hP3 | \(P3_{1}21\) | 4.466 | – | 5.919 | – | \(\gamma -{\rm Se}\) | [26] |
Thermodynamic data
Phase(s) | Quantity | Method(s) | T [K] | at.% Te | References |
---|---|---|---|---|---|
Liquid | \(a_{\text {Te}}\) | Isopiestic | 1170–1207 | 46.1–51.8 | [14] |
\(H_{\text {Ni}}^{E,\infty }\) | Calorimeter | 741 | 99.5 | [40] | |
\(\beta 2\) | \(C_{\text {P}}\) | AShC, DSC | 218–958 | 40–41.2 | |
\(\Delta H_{\text {for}} ^{298}\) | Sol. Cal. | 298 | 40.5 | [32] | |
\(\mu _{\text {Ni}}\) | EMF | 963–1050 | 40.5 | [39] | |
\(P_{\text {Te}_{2}}\) | KEMS | 981–1039 | 40.5 | [35] | |
\(\beta 1\) | \(\mu _{\text {Ni}}\) | EMF | 1080–1178 | 40.5 | [39] |
\(a_{\text {Te}}\) | Isopiestic | 1106–1236 | 38.7–42.5 | [14] | |
\(P_{\text {Te}_{2}}\) | KEMS | 1036–1150 | 39 and 40.5 | [35] | |
\(\delta \) | \(C_{\text {P}}\) | Cryo, AShC, AScC | 5–930 | 52.3–66.7 | |
\(\Delta H_{\text {for}} ^{0 \, K}\) | DFT | 0 | 50 and 66.67 | [34] | |
\(a_{\text {Te}}\) | Isopiestic | 765–1178 | 52.5–66.3 | [14] | |
\(\mu _{\text {Ni}}\) | EMF | 673 and 700 | 54.4–66.7 | ||
Various 2-phase | \(a_\mathrm{Te}\) | Isopiestic | 759–1252 | 34.9–70.1 | [14] |
\(\gamma 2+\delta \) | \(a_{\text {Te}}\) | Isopiestic | 1084–1126 | 51.2–52.4 | [14] |
\(\delta +\gamma 1\) | \(\Delta H_{\text {for}} ^{298}\) | Sol. Cal. | 298 | 50 | [33] |
\(\mu _{\text {Ni}}\) | EMF | 673 and 700 | 43.5–52.6 | ||
\(\gamma -\mathrm{Ni}+\beta 1\) | \(P_{\text {Te}_{2}}\) | KEMS | 1020–1190 | 25–35 | [36] |
\(\gamma -\mathrm{Ni}+\beta 2\) | \(P_{\text {Te}_{2}}\) | KEMS | 893–993 | 25–35 | [36] |
DFT methodology
Converged | ||||||||
---|---|---|---|---|---|---|---|---|
Valence | \(E_\mathrm{cutoff}\) | Converged | \(V_\mathrm{rel}\) [Å] | \(K_{V}\) | \(K_{\rm exp}\) | |||
Element | Stable phase | Potential used | \(e^{-}\) | [eV] | K-points | (ref) | [GPa] | [GPa] |
Ni | FCC_A1 | Ni_sv_GW 2013 (GGA) | \(3d^{9}4s^{1}\) | 570 | 20 × 20 × 20 | 43.4 (43.8 [51]) | 208.3 | 185.3 [49] |
Te | A8 hexagonal | GGA_GW 2012 | \(5s^{2}5p^{4}\) | 330 | 19 × 19 × 15 | 104.9 (101.8 [26]) | 20.3 | 27.34 [50] |
Calphad methodology
Solution phases
Table 4 summarizes the sublattice models of all intermediate phases used in this assessment of the Ni–Te system, with their respective end-members and composition ranges.
For simplicity, the \(\beta 1\) phase was modeled with two sublattices as \((\mathrm{Ni,Te})_{2}(\mathrm{Te})_{1}\), since the crystal structure is not well known. The 2:1 ratio was chosen to create an end-member close to the Ni-rich phase boundary. The order–disorder transformation into \(\beta 1'\) at lower temperature and higher Ni content is disregarded in this work. The \(\beta 2\) phase was modeled as \((\mathrm{Ni})_{1}(\mathrm{Te})_{1}(\mathrm{Ni,Va})_{1}\) according to the tetragonal P4 / nmm space group, compatible to the Fe–Te \(\beta \) phase modeled in a previous assessment [8]. The low-temperature ordering into space groups P21 / m (monoclinic) and Pma2 (orthorhombic) at high and low Ni content, respectively, is not modeled here since it is not of particular interest to the application of nuclear reactors, which operate at higher temperatures. The \(\delta \) phase was modeled as \((\mathrm{Ni,Va})_{1}(\mathrm{Ni,Va})_{1}(\mathrm{Te})_{2}\) in order to facilitate the second-order transition between the NiAs type disordering of vacancies to the CdI\(_{2}\)-type layering of vacancies in every other Ni layer (hence the separation of the (Ni,Va) interstitial sites into two sublattices) as has been experimentally verified [18, 22]. This is how the \(\delta \) phase was previously modeled in the Fe–Te system [8], and compatibility was accommodated for complete exchange of Fe and Ni in the Fe–Ni–Te system.
Liquid phase
Sublattice models used for alloy phases of the Ni–Te system
Phase | Sublattice model | End-members | Range at.% Te | Comment |
---|---|---|---|---|
Liquid | \((\mathrm{Ni}^{+2})_\text {P}(\mathrm{Te}^{-2},Va^{-Q},\mathrm{Te}^{0})_\text {Q}\) | Ni, \(\mathrm{Ni}_{2}\mathrm{Te}_{2}\),Te | 0–100 | Ionic liquid |
\(\text {BCC}\) | \((\mathrm{Ni,Te})_{1}(\mathrm{Va})_{3}\) | \(\mathrm{Ni}_{1}\mathrm{Va}_{3}\), \(\mathrm{Te}_{1}\mathrm{Va}_{3}\) | 0–100 | Ideal solution |
\(\text {FCC}\) | \((\mathrm{Ni,Te})_{1}(\mathrm{Va})_{1}\) | \(\mathrm{Ni}_{1}\mathrm{Va}_{1}\), \(\mathrm{Te}_{1}\mathrm{Va}_{1}\) | 0–100 | Ideal solution |
\(\text {HCP}\) | \((\mathrm{Ni,Te})_{1}(\mathrm{Va})_{0.5}\) | \(\mathrm{Ni}_{1}\mathrm{Va}_{0.5}\), \(\mathrm{Te}_{1}\mathrm{Va}_{0.5}\) | 0–100 | Ideal solution |
\(\beta 1\) | \((\mathrm{Ni,Te})_{2}(\mathrm{Te})_{1}\) | \(\mathrm{Ni}_{2}\mathrm{Te}_{1}\), \(\mathrm{Te}_{2}\mathrm{Te}_{1}\) | 33.33–100 | |
\(\beta 2\) | \((\mathrm{Ni})_{1}(\mathrm{Ni,Va})_{1}(\mathrm{Te})_{1}\) | \(\mathrm{Ni}_{1}\mathrm{Ni}_{1}\mathrm{Te}_{1}\), \(\mathrm{Ni}_{1}\mathrm{Va}_{1}\mathrm{Te}_{1}\) | 33.33–50 | |
\(\gamma 1\) | \((\mathrm{Ni})_{52}(\mathrm{Te})_{40}\) | \(\mathrm{Ni}_{52}\mathrm{Te}_{40}\) | 43.48 | |
\(\gamma 2\) | \((\mathrm{Ni})_{20}(\mathrm{Te})_{17}\) | \(\mathrm{Ni}_{20}\mathrm{Te}_{17}\) | 45.95 | |
\(\delta \) | \((\mathrm{Ni,Va})_{1}(\mathrm{Ni,Va})_{1}(\mathrm{Te})_{2}\) | \(\mathrm{Ni}_{1}\mathrm{Ni}_{1}\mathrm{Te}_{2}\), \(\mathrm{Ni}_{1}\mathrm{Va}_{1}\mathrm{Te}_{2}\)\(\mathrm{Va}_{1}\mathrm{Ni}_{1}\mathrm{Te}_{2}\), \(\mathrm{Va}_{1}\mathrm{Va}_{1}\mathrm{Te}_{2}\) | 50–100 | \(\mathrm{Ni}_{1}\mathrm{Va}_{1}\mathrm{Te}_{2}\) and \(\mathrm{Va}_{1}\mathrm{Ni}_{1}\mathrm{Te}_{2}\), are equivalent |
Stoichiometric phases
The stoichiometric \(\gamma 1\) phase was modeled as \((\mathrm Ni)_{52}(\mathrm Te)_{40}\) and the \(\gamma 2\) phase as \((\mathrm Ni)_{20}(\mathrm Te)_{17}\), and the NKR was used for both. The pure Te phase (hexagonal A8) and \(\gamma -\)Ni (FCC), \(\alpha -\)Ni (BCC) and \(\epsilon -\)Ni (HCP) were taken directly from Dinsdale’s descriptions in the PURE5 database from SGTE [55]. No nickel was added to the A8 phase, but Te was added to the Ni sites of FCC, BCC and HCP. HCP was included in the assessment only to enter the tellurium lattice stability evaluated via DFT.
Optimization procedure
A thermodynamic assessment was performed using the PARROT module of the Thermo-Calc Software package [56] for parameter optimization. The assessment began by using a substitutional solution model for the liquid, with parameters of appropriate order of magnitude. Thereafter, solid phases were introduced one by one and optimized in the following manner.
Liquid, gas and terminal phases
The gas phase description was not optimized in this work, but merely extracted from the SGTE SSUB5 database [57, 58, 59, 60].
After solid phases had been introduced, liquid model parameters were optimized to fit the congruent points of the \(\beta 1\) and \(\delta \) phases, as well as a tentative liquidus point representing the “melting effect” found at 1663 K for 15 at.% Te by Klepp and Komarek [15]. There were no further details about that data point in their paper. When those solid phases were approximately in place, the interaction terms in the liquid could be further optimized to fit liquidus and activity data [11, 14, 15], and the liquid \(\mathrm{Ni}_{2}\mathrm{Te}_{2}\) end-member was used to help fit the eutectic point between \(\gamma 2\) and \(\delta \).
No solubility was modeled in the pure terminal phases, although a small amount of Ni might be soluble in pure Te [24].
\(\beta 1\) and \(\delta \) phases
The end-member and interaction parameters of the \(\beta 1\) phase were first manually adjusted in order to get the phase in the approximately correct location in the phase diagram. The parameters were then optimized to fit the \(L\rightarrow \mathrm{FCC}+\beta 1\) eutecticum and activity data [14, 35, 36, 39]
All terms except the entropy terms (bT in Eq. 6) of the \(\delta -\)NiTe and \(\delta -\mathrm{NiTe}_{2}\) end-members were fitted to the formation enthalpy data evaluated via DFT by Jandl et al. [34] and the heat capacity data of Grønvold [29] and Tsuji and Ishida [31]. The interaction terms were then optimized to fit activity data [14, 21, 38] and the bT terms to fit solidus data [14, 15] and the phase boundaries evaluated by Barstad et al. [13]. The interaction term \(^{0}L^{\delta }_{\mathrm{Ni,Va:Va:Te}}={^{0}}L^{\delta }_{\mathrm{Va:Ni,Va:Te}}\), affecting the range NiTe\(_{2}-\)Te, was given a constant positive value in order to suppress a higher Te-solubility than about 66.67 at.% Te.
\(\beta 2\), \(\gamma 1\) and \(\gamma 2\) phases
The end-members of the \(\beta 2\) phase were first manually adjusted to make the phase appear in approximately the correct temperature and composition range. Both end-members, i.e., \(\beta 2-\mathrm{Ni}_{2}Te\) and \(\beta 2-\)NiTe, were modeled using the NKR. The stable composition range of the phase is narrow and lies roughly in the middle between the end-members; the end-members are thus rather far away from the compositions of available heat capacity data [29], which made it difficult to fit the data with end-members described as power series (Eq. 6). Instead, it worked well to model it as a power-series contribution to the interaction parameter \(^{0}L^{\beta 2}_{\mathrm{Ni:Ni,Va:Te}}\). The a- and b-terms of the interaction parameters together with the b-term of \(\beta 2-\mathrm{Ni}_{2}\mathrm{Te}\) could then be optimized to fit activity data [35, 39] and the congruent reaction \(\beta 2 \rightleftharpoons \beta 1\). The a-terms of the end-members were optimized to fit a sort of compromise between the enthalpy of formation of the end-members evaluated via DFT in this work and the calorimetric point by Shukla et al. [32]. If the latter enthalpy of formation were to be fit well, the phase would be too stable to allow the proper shape of the \(\delta \) phase.
The a-term of the \(\gamma 1\) phase was first fitted to the DFT enthalpy of formation value evaluated in this work, but was then lowered to fit the phase diagram; with a formation enthalpy of the \(\beta 2\) phase so low, the \(\gamma 1\) phase enthalpy must also be rather low to form the Gibbs energy tangent between the \(\beta 2\) and \(\delta \) necessary to have them all stable at low T. The b-term was fitted to relevant invariant reactions. The \(\gamma 2\) phase was added last, and its parameters were optimized to fit the phase diagram only, since there are virtually no other data available on the phase.
DFT Results and discussion
Relaxed lattice parameters and 0 K formation energies (\(\Delta E_\text {f}\)) from DFT calculations of Ni–Te end-members. [17]
Phase | \(E_\text {cutoff}\) | k-points | Relaxed lattice parameters [Å] | \(\Delta E_\text {f}\) | ||
---|---|---|---|---|---|---|
[eV] | a | b | c | [kJ/mol] | ||
\(\beta 2-\mathrm{Ni}_{2}\mathrm{Te}\) | 570 | 15 × 15 × 10 | 3.88 | – | 6.02 | − 7.8 |
\(\beta 2-\mathrm{NiTe}\) | 570 | 15 × 15 × 10 | 3.75 | – | 6.03 | − 16.8 |
Exp. \(\beta 2-\mathrm{Ni}_{}\) [17] | 3.78 | 6.06 | ||||
\(\gamma 1-\mathrm{Ni}_{52}\mathrm{Te}_{40}\) | 430 | 3 × 2 × 6 | Volume not relaxed | − 16.7 | ||
\(\text {HCP\_A3}-Te\) | 370 | 31 × 31 × 19 | 4.06 | – | 4.39 | 17.3 |
Calphad modeling results and discussion
Thermodynamic properties
The formation enthalpy by Jandl et al. is very close to the calorimetric data point of a NiTe sample by Predel and Ruge [33], whose analyzed composition corresponds to a two-phase equilibrium of \(\delta +\gamma 1.\) Figure 3 shows the calculated enthalpy of formation, separately calculated, for all phases in the system. The enthalpy of the \(\beta 2\) phase closer fits DFT values than the experimental data point, while the \(\delta \) phase has a lower enthalpy than predicted by DFT at the NiTe end-member [34]. Therefore, the \(\gamma 1\) phase has a lower enthalpy than predicted by DFT in order to lie on a line in Gibbs energy between \(\beta 2\) and \(\delta. \)
Figure 4 shows the calculated heat capacity of \(\beta 2\) compared with experimental data of 40 at.% Te and 41.1 at.% Te [28]. The lambda peaks were not modeled; it is a satisfying fit to the pseudo-linear portions of the datasets. The heat capacity of the \(\delta \) phase fits experimental data well [27, 29, 31], as shown in Figs. 5 and 6. Figure 6 shows for 57.1 at.% Te a bump in \(C_\mathrm{P}\) at around 350 K; this was not modeled but the general slope of the data is satisfactory.
Figures 7 and 8 show the negative logarithm of the calculated thermodynamic activity of the Ni–Te system compared with the isopiestic data by Ettenberg [14] at 600 and \({875}\,^\circ \hbox {C}\), respectively. It was difficult to obtain a good fit together with other activity data (as seen in Figs. 9,10, 11 and 12), and the greater deviation from experiments in Fig. 8 is due to the Te-rich solidus of the \(\delta \) phase not matching the experimental phase diagram data [11, 15] at that temperature, as will be presented in "Phase diagram" section. Note that the chemical potential in Fig. 10 seems to deviate much from experimental data, but that the scale of the maximum deviation is only about 6%; it was difficult to improve the fit beyond this.
Phase diagram
The calculated phase diagram of the Ni–Te system is shown in Fig. 13. The final description is a compromise of fitting thermodynamic data and the phase diagram as best as reasonably achievable, and it is apparent from the figure that some aspects have been sacrificed. As seen in the zoomed-in Fig. 14 the \(\beta 1\) and \(\beta 2,\) congruent and solvus lines do not fit perfectly, whereas the invariant reactions are well reproduced. The \(\beta 1\) liquidus on the Te-rich side is slightly higher than measured by Klepp and Komarek [15] while the solvus fits well.
As a result of the optimization, a transition between NiAs-type disorder of interstitials and \(CdI_{2}\) order appears in the \(\delta \) phase. This transition was calculated for the metastable \(\delta \) phase diagram and overlies the zoomed-in Ni–Te diagram in Fig. 15. The position of this line could be optimized to fit the data in the literature [18, 22], but doing so resulted in a poor fit of the phase boundaries; therefore, it was not prioritized with the application in mind. Figure 16 shows the site fractions of all constituents in the \(\delta \) phase from 50 to 66.67 at.% Te at 962 K. It is seen that the interstitials are fully disordered (NiAs-like) up to 52.06 at.% Te, i.e., outside the equilibrium solubility limit of Ni in the phase at 52.17 at.% Te, above which nickel rapidly shifts to the second sublattice; therefore, the completely disordered region only exists mainly in the metastable delta phase region, although there is a very narrow stable disordered interval right by the leftmost \(\delta \) corner at the invariant \( \gamma 2\leftrightarrow \gamma 1+\delta \). At 54.3 at.% Te, the phase is very close to \(CdI_{2}\) type ordering with \(y'' _\mathrm{Ni}=0.99\).
Conclusions and future work
A thermodynamic assessment of the Ni–Te system has been performed, supported by DFT calculations. The thermodynamic description is deemed good for the conditions present in the application to nuclear reactors. The liquid was modeled using the ionic two-sublattice model. There is an order–disorder transition in the metastable \(\delta \) phase but its exact position was not optimized. The description does not model all known ordered superstructures of the Ni–Te alloys; for such applications where that is desired, the description should be modified to include those phases, e.g., by modifying the sublattice models accordingly, introducing ordering parameters, or modeling them as separate phases. An assessment of the Fe–Ni–Te system is in progress.
Notes
Acknowledgements
The authors are grateful to the Swedish Research Council (Vetenskapsrådet) for funding the SAFARI project. Carl-Magnus thanks his colleagues at CEA and KTH, as well as Drs Bonnie Lindahl and Nathalie Dupin for invaluable advice on modeling, Andrei Ruban for advice about DFT calculations. C.-M. further extends thanks to Mats Kronberg at the National Supercomputer Centre, Linköping, for keeping a backup of the DFT data when it was believed to be erased.
Compliance with ethical standards
Conflicts of interest
The authors declare that they have no conflict of interest.
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