Extreme structural stability of Ti 0.5 Sn 0.5 O 2 nanoparticles: synergistic effect in the cationic sublattice † Denis Machon, * a,b Sylvie Le Floch, a Shashank Mishra, c Stéphane Daniele, c Karine Masenelli-Varlot, d Patrick Hermet e and Patrice Mélinon a Ti 0.5 Sn 0.5 O 2 nanoparticles (∼5 nm and ∼10 nm) have been studied under high pressure by Raman spec-troscopy. For particles with diameter ∼10 nm, a transformation has been observed at 20-25 GPa while for particles with ∼5 nm diameter no phase transition has been observed up to ∼30 GPa. The Ti 0.5 Sn 0.5 O 2 solid solution shows an extended stability at the nanoscale, both of its cationic and anionic sublattices. This ultrastability originates from the contribution of Ti and Sn mixing: Sn stabilizes the cationic network at high pressure and Ti ensures a coupling between the cationic and anionic sublattices. This result ques-tions a "traditional" crystallographic description based on polyhedra packing and this synergistic effect reported in this work is similar to the case of metamaterials but at the nanoscale.

At the nanoscale, the phase stability of a system is often modified compared to the bulk. The thermodynamics and kinetics of phase transitions are strongly impacted by the contribution of the interface energy and some additional size effects (confinements, molecular effect, etc.). 1 For example, under ambient conditions, polymorphs that differ from the bulk counterpart can be stabilized in nanostructures. This is a typical size-induced phase transition, as reported for Ni 2 or for TiO 2 . 3 In the latter case, the anatase-type structure (space group #141, I4 1 /amd )i so b s e r v e di n nanoparticles while a rutile-type structure (space group #136, P4 2 / mnm) is the stable polymorph for the bulk. Likewise, temperatureor pressure-induced phase transitions in nanomaterials often show significant differences with the bulk. For example, the melting temperature of a compound appeared to depend on the inverse radius of the particles. 4,5 More recently, pressure-induced phase transformations have attracted particular interest because, unlike some high temperature experiments, particle size is not affected by growth and coarsening. The seminal works of Tolbert and Alivisatos showed that the transition pressure can also vary a sa1 / R-law in some cases [ref. 6 and references therein]. However, more recent works demonstrated that the situation can be more complex depending on the system under investigation. Under pressure, change in transition pressure or high-pressure structures, and sometimes amorphization can be observed. 1,[7][8][9] For instance, with increasing pressure, bulk rutile SnO 2 undergoes the following sequence of structural transitions: rutile-type → CaCl 2 -type → cubic phase (modified fluorite, space group #205, Pa3 ˉ). 10 I nt h en a n o p o w d e r ,t h eC a C l 2 -type phase (space group #58, Pnnm) has not been observed and the transformation to the high-pressure cubic phase is strongly dependent on the particle size. 11 The cubic structure appeared above 23 GPa, 29 GPa and 30 GPa for bulk, 14 nm and 8 nm SnO 2 samples, respectively. For a particle diameter of 3 nm, no phase transition could b eo b s e r v e da tp r e s s u r e sa sh i g ha s3 9G P a .H o w e v e r ,R a m a n spectroscopy measurements have shown that a pressure-induced disordering of the anionic sublattice starts above 7 GPa in 3 nm diameter particles. 12 Regarding TiO 2 , whatever the initial structure (anatase or rutile), a baddeleyite structure (space group #14, P2 1 /c) has been observed at high pressure, above ∼18 GPa. 13,14 Depending on the density of defects in nanoparticles, a highdensity amorphous state could be found in place of the baddeleyite structure upon compression. [15][16][17][18] In materials science, another degree of freedom in the Gibbs' phase rule is composition. This led to explore and develop alloys or mixed oxides. In this case as well, size and ∼1700 K. However, in nanoparticles, such demixing may be bypassed, leading to an extended stability field of the solid solution. 19 Here, we want to take advantage of this compositional stability at ambient temperature to study the pressure-induced behavior of the Ti 0.5 Sn 0.5 O 2 solid solution, and to explore the joint effect of pressure and composition in the energy landscape of this system. Using two different particle sizes (∼5 and ∼10 nm), we demonstrate the size dependence of the transition pressure. The combination of different parameters (size and pressure) allows us to explore the stability fields of the solid solution. In the case of 5 nm particle, the mechanical stability is extremely large compared to the end members. In addition, no decoupling between the cationic and anionic is observed unlike in SnO 2 . The high-pressure behavior of these ultrafine nanoparticles allows discussing the respective role of each cation and their synergistic effect and challenge the universality of the traditional crystallographic description based on Pauling's rules. 20 scheme. 23 Relaxation of the lattice parameters and the atomic positions was performed using a Broyden-Fletcher-Goldfarb-Shanno algorithm until the maximal residual force and pressure were less than 5 × 10 -6 Ha Bohr -1 and 1 × 10 -3 GPa, respectively. At the end of this structure relaxation, the calculated lattice parameters, a = 4.65 Å and c = 3.05 Å, were in excellent agreement with the experimental ones (see next section). Dynamical matrix, dielectric constants and Born effective charges are calculated within a variational approach to density functional perturbation theory (DFPT). Phonon dispersion curves were interpolated according to the scheme described by Gonze et al. 24 The dipole-dipole interactions were subtracted from the dynamical matrices before Fourier transformation, so that only the short-range part was handled in real space. We considered a 2×2×2q-points grid for the calculation of the phonon band structure and a denser 100 × 100 × 100 grid was used for the calculation of the phonon density-of-states.

Experimental

Raman spectroscopy at high pressure Raman experiments were carried out using a Horiba low frequency Labram Raman spectrometer operated with a 532 nm wavelength laser compatible with our high-pressure setup (Diamond Anvil Cell -DAC), which can detect an inelastic signal down to about 6 cm -1 . Laser power was set at 5 mW at the entrance of the DAC to avoid heating. The beam was focused on the sample using a 50× objective, with beam diameter ∼2 μm at the sample. The scattered light was collected in backscattering geometry using the same objective.

High pressure was generated using a membrane DAC with low-fluorescence diamonds. Samples were placed into a 125 μm chamber drilled in an indented stainless-steel gasket. No pressure transmitting medium was used. The pressure was probed by the shift of the R1 fluorescence line of a small ruby chip.

Synthesis and characterization of Ti 0.5 Sn 0.5 O 2 nanoparticles at ambient conditions Synthesis of Ti 0.5 Sn 0.5 O 2 nanoparticles has been described elsewhere. 25 In brief, 10 g of [SnCl 4 Ti(OEt) 4 (EtOH) 2 ] were added to 100 mL of boiling water and stirred for 2 h. Addition of NH 4 OH until pH 6-7 was reached led to the precipitation of a white powder which could be recovered by centrifugation, washed twice with de-ionized water and ethanol and finally dried at 70 °C for 12 h (4 g). The as-prepared particles were divided into two batches that were annealed following two different conditions, at 750 K during 4 h (batch 1) and at 1120 K during 2 h (batch 2). These temperatures have been chosen to obtain batches with significantly different sizes. Higher temperatures are prohibited as a spark plasma sintering treatment at 1370 K led to a demixing. 25 In the latter case, clear signature of appearance of TiO 2 and SnO 2 was reported interfaces play a significant role in the modification of the energy landscape. For instance, the spinodal decomposition observed in some bulk alloys may be bypassed for nanoparticles with radii smaller than a critical value. For example, in the TiO 2 -SnO 2 system, the solid solution is stable at high temperature but equimolar systems of TiO 2 and SnO 2 shows demixing to Ti and Sn-rich solids at temperatures below

Methods

Simulations

First-principles based methods were performed within the density functional theory (DFT) framework as implemented in the ABINIT package. 21 The exchange-correlation energy functional was evaluated using the local density approximation parametrized by Perdew and Wang. 22 The all-electron potentials were replaced by norm-conserving pseudopotentials. Sn (4d 10 ,5s 2 ,5p 2 ), Ti(3s 2 ,3p 6 ,3d 2 ,4s 2 ) and O(2s 2 ,2p 4 )-electrons were considered as valence states. In the case of tin atoms, there is no reason to introduce semi-core electrons because no calculation has shown the importance of introducing semicores in a compound. In contrast, the semi-cores (3s 2 and 3p 6 ) have to be introduced in the construction of the Ti-peudopotential to have reliable results. With semi-core electrons, the calculated structure of TiO 2 is in better agreement with the experimental data than without semi-core electrons. Thus, the semi-core electrons for titanium are mandatory for the calculation of Ti 0.5 Sn 0.5 O 2 and the results can be compared with the calculated vibrational density-of-states previously published for TiO 2 nanorods. 14 Therefore, nothing in the vibrational or stability properties indicates that a valence transfer or fluctuation should be considered.

The electronic wave functions were expanded in planewaves up to a kinetic energy cutoff of 65 Ha and integrals over the Brillouin zone were approximated by sums over a 10 × 10 × 8 mesh of special k-points according to the Monkhorst-Pack by TEM and XRD. 25 In our present case, no trace of endmembers has been observed.

According to X-Ray diffraction (Fig. 1a), and considering the nanoparticles are single-domain as verified by TEM (see below), the Scherrer equation indicates that the particle size of batch 1 i sc e n t e r e do n4±1n mw h e r e a sb a t c h2h a sap a r t i c l es i z e around 12 ± 1 nm. TEM analysis on this batch (Fig. 1b andc) allows obtaining the particle size distribution (see Fig. S1-3 †) and to refine the particle size. It gives an average size of 5 ± 1 nm and 10 ± 2 nm for batch 1 and batch 2, respectively.

In both cases, the indexations of the X-ray diffraction pattern are in agreement with a rutile structure with cell parameters a = 4.67(1) Å and c = 3.07(1) Å. This corresponds to the expected parameters predicted by the Vegar'sl a wi.e., a linear combination of the end member cell parameters when considering a substitutional solid solution. 26,27 This observation is an indication that there is no demixing and no surface segregation as observed in some cases. 28 This difference may originate from the synthesis process using a single source precursor (SSP) and the different situations found in binary nanoparticles (solid solution, janus, core-shell). 29,30 The Raman spectrum of batch 1 is shown in Fig. 2 along with spectra of end-member oxides, i.e., TiO 2 and SnO 2 , either as bulk (b) or nanostructure (n). It is worth noting that TiO 2 nanoparticles usually crystallizes in the anatase structure and no reference Raman spectrum of the rutile phase could be obtained with ultrafine nanoparticles. The reference spectrum for nanostructured rutile TiO 2 is therefore extracted from ref.

14, as nanorods with 8 nm-diameter were found to remain in the rutile structure.

The analysis of the vibrational representation of the normal modes at the center of the Brillouin zone 31 indicates that four Raman-active modes are expected for the rutile-type structure (A 1g ,B 1g ,B 2g ,E g ). It has to be noted that in TiO 2 , the B 2g mode is very weak. In SnO 2 , the B 2g mode is visible but not the B 1g one, as supported by calculations. 32 For all these vibrational modes, the oxygen atoms vibrate while cations are at rest. As already discussed for TiO 2 nanorods, the large band at ∼120 cm -1 is not expected according to the Raman selection rules but is originated by defects. 14 Indeed, the Raman selection rules are established for an infinite perfect crystal. The presence of defects (usually oxygen vacancies in oxides 33 ) breaks the usual selection rules and gives access to forbidden vibrational modes and also to k ≠ 0 modes, i.e., modes in the Brillouin zone. Such effect is revealed by comparing the Raman spectra with the calculated vibrational density of states (vDOS) of the solid solution (Fig. 2b). The correspondence between the Raman peaks and the maxima of the simulated vDOS of the mixed system also confirms that there is no segregation i.e., demixting between Ti-rich and Sn-rich compounds.

Raman spectra from both batches of nanoparticles are compared in Fig. 3. There is no strong difference except at low-frequency, below 40 cm -1 . In that spectral range, there is a defined peak in the case of batch 1. This peak located at ∼20 cm -1 corresponds to the confined acoustic vibration in nanoparticles [ref. 36 and references therein]. As the frequency is inversely proportional to the radius of the particles in the Lamb's model, 37 this peak is hardly observed for batch 2 under ambient conditions. For this particle size (10 nm), the expected peak position should be located around ∼8c m -1 , which is the detection limit of this peak using this experimental configuration. This peak is hardly observable at ambient pressure but better observed with increasing pressure, as described in the following section. It is worth noting that the observation of these low-frequency peaks is an indication that the nanoparticles are free standing (in agreement with the TEM observations). They would not be observable if there was some sintering of the nanoparticles.

Results

Ti 0.5 Sn 0.5 O 2 -5nm Raman spectra of 5 nm diameter nanoparticles (batch 1) under pressure are shown in Fig. 4. No phase transition can be observed up to 30.1 GPa, the highest pressure achieved in this run. It is worth noting that such extended stability is surprising as both TiO 2 and SnO 2 , bulk or nanostructures, undergo phase transitions below 20 GPa. In the case of TiO 2 nanorods, phase transition towards a monoclinic phase (baddeleyite) occurs at ∼16 GPa. 14 A similar transformation is observed for TiO 2 nanoparticles with anatase as the initial structure. 13,18 In SnO 2 nanoparticles, a pressure-induced disordering has been reported starting above 7 GPa with a disorder propagation from the shell to the core ending in the anionic sublattice amorphization above 19 GPa. 12 Fig. 3 Comparison of Raman spectra of batch 1 (mean particle size: 5 nm) and batch 2 (mean particle size: 10 nm). (a) Frequencies up to 900 cm -1 . (b) Zoom on frequencies below 60 cm -1 .

One observes that the spectrum of Ti 0.5 Sn 0.5 O 2 nanoparticles is very similar to that of TiO 2 in its rutile phase and rather different from nanometric rutile SnO 2 . It is well documented that the Raman spectra of SnO 2 are strongly affected by the nanometrization with a strong broadening of the Raman peaks, 34,35 as it can be seen by comparing n-SnO 2 and b-SnO 2 in Fig. 2a. This effect can be discussed as resulting from the effect of defects on the decoupled anionic and cationic sublattices. 32 Thus, the interesting observation of the similarity between the Ti 0.5 Sn 0.5 O 2 and TiO 2 Raman spectra indicates that decoupling between sublattices is not observed in the solid solution (as it is in the case of SnO 2 nanoparticles) despite the presence of defects.

The pressure dependences of the Raman modes are indicated in Table 1 for both batches and for TiO 2 and SnO 2 nanostructures. The pressure-dependencies are closer to those observed in n-TiO 2 than in n-SnO 2 . This seems to indicate a bond energy dominated by the Ti-O interaction.

The low-frequency peak (confined acoustic mode) can be clearly monitored above 3 GPa and shows a non-linear behavior with increasing pressure. This pressure-induced evolution (Fig. S4 †) can be roughly divided into two regimes with increasing pressure. The first regime, up to ∼12 GPa, can be approximated by a linear variation with pressure with a slope of 0.8 cm -1 GPa -1 . This value is close to the observed variation in SnO 2 nanoparticles but it should be noted that this value is not available for TiO 2 rutile nanoparticles. The second regime shows a drastic reduction of this variation and the peak position is apparently unaffected by the pressure application. This is related to the pressure-dependency of the sound velocity, as discussed in ref. 38.

The decompression shows a totally reversible path with a Raman spectrum after pressure cycling very similar to the Raman spectrum before compression (Fig. 5). The compaction induces a narrowing of the Rayleigh line allowing a better definition of the low-frequency Raman peak. Ti 0.5 Sn 0.5 O 2 -10 nm Using the same experimental set-up, high-pressure experiment was performed on batch 2. The Raman spectra during compression are shown in Fig. 6 and the pressure-dependencies of the peak positions are reported in Table 1. They are similar to that observed for batch 1. The low-frequency mode also shows a similar behavior (Fig. S5 †). However, contrary to the previous experiment on batch 1, i.e. on smaller nanoparticles, a phase transition is observed between 20.3 and 25.1 GPa. It is worth noting that the low-frequency peak disappears at the transition confirming that this signal is clearly related to the sample and its low-pressure structure. The high-pressure phase is difficult to identify because of the broadness of the peaks. It may correspond to a highly disordered baddeleyite structure, as observed in pure TiO 2 nanostructures. 14 One may also interpret it as amorphous. However, on decompression, a crystalline structure is observed (Fig. 7). Usually, in TiO 2 , the reported pressure-induced amorphization is irreversible. 17 Therefore, the high-pressure phase is probably not amorphous but highly disordered. It may result from a frustration in the transformation of the anionic sublattice. As a matter of fact, the high-pressure phases of TiO 2 and SnO 2 (modified fluorite) exhibit a distorted fcc cationic sublattice but the anionic arrays differ, leading to a metal coordination number of 7 in the baddeleyite structure (TiO 2 ) and 6 + 2 in the modified fluorite one (SnO 2 ). 39 On decompression, the Raman spectra show some changes that are significant but are difficult to interpret because of the Table 1 Pressure-derivatives of the peak positions for the Ti 0.5 Sn 0.5 O 2 nanoparticles (batches 1 and 2) compared to those of SnO 2 12 and TiO 2 . 14 The label "disorder" corresponds to the band observed at 120 cm -1 (see text) anatase and the baddeleyite phases. In nanostructures, this phase is bypassed during compression but is generally observed on decompression (Fig. S6 †). 13,18 This phase is also observed in SnO 2 under pressure, but under non-hydrostatic broadness of the Raman bands. Below 6.4 GPa a typical spectrum of the columbite (or α-PbO 2 ) structure (space group #60, Pbcn) is obtained. This orthorhombic structure is usually observed on bulk TiO 2 under increasing pressure, between the the difference in interfacial energy between the low-and highpressure phases. This is the case of SnO 2 nanoparticles: a pressure-induced phase transition to the high-pressure cubic phase is observed at 30 GPa for an 8 nm-particle while this transition pressure is 23 GPa for bulk SnO 2 .F o r3n mp a r t i c l e s ,n o phase transition is observed up to at least 39 GPa by X-ray diffraction techniques and the phase remains the rutile structure. 11 In TiO 2 , no such clear 1/R dependence has been reported. The kinetic effect can be described by the classical nucleation theory, 41 which shows that the nucleation of a phase leads to a gain of volumetric energy but at a cost of a surface energy. Therefore, the radius of a critical nucleus can be calculated and is usually on the order of a few nanometers. If this critical diameter is larger than the size of the nanoparticles, the nucleation of the new phase can be bypassed. In the case of anatase TiO 2 , a kinetic effect has been proposed to explain the bypass of the intermediate columbite phase. 13 Thus, it has been shown that the pressure-induced transformation from anatase to columbite (α-PbO 2 ) requires a nucleus of ∼12 nm. The critical size of the nucleus is ∼4 nm for the next highpressure baddeleyite phase, 13 the one that is usually observed in nanomaterials of critical size (diameter of the nanoparticle or diameter of the rods) less than 10 nm (ref. 13, 14, 16 and 18) and with a low density of defects. 13,18 In the case of Ti 0.5 Sn 0.5 O 2 , the observation of a pressureinduced phase transition for the 10 nm nanoparticles above ∼20 GPa but no phase transition for those of 5 nm, at least up to ∼30 GPa, indicates that the trends is similar to SnO 2 with a size dependence on the transition pressure, a situation not observed in nanostructured TiO 2 . conditions, and after a pressure cycle. 39 The α-PbO 2 phase is the kinetically preferred phase during the decompression and is retained as a metastable phase. This is due to orientational relationships that facilitate certain transition pathway as discussed in ref. 39.

Symmetry

In conclusion, for nanoparticles with a diameter centered at ∼10 nm, an abrupt phase transition is observed above 20 GPa without preliminary disordering. The transformation to the high-pressure structure occurred above ∼20 GPa in Ti 0.5 Sn 0.5 O 2 vs. ∼16-18 GPa for the TiO 2 rutile nanorods or anatase nanoparticles. 13,14,18 On decompression, the recovered phase is the α-PbO 2 structure.

Discussion

In view of the experimental results reported in this work, there are at least two effects inducing an extended range of stability of the rutile phase: (i) a size effect as it clearly appears that the size plays a significant role in the phase transition and the stability field of the different phases involved; (2) the impact of the solid solution with respect to the end-members (TiO 2 and SnO 2 ).

First, the increase of the transition pressure with decreasing size is a general effect and has been documented. Generally, it results from two considerations (and their combination): kinetics and thermodynamics. Recent works has formalized the increase in stability at the nanoscale either by a classical (Gibbs) approach or using the Landau theory of phase transitions. 40 This is a 1/R scaling law showing that the transition pressure is shifted to a higher value with decreasing size and depending on brought to date. [START_REF] Vegas | Structural models of inorganic crystals: From the elements to the compounds[END_REF] The traditional crystallographic description of the structure of ionic crystals relies on the packing of fundamental building blocks generally considered to be a cationcentered polyhedron. This description is generally associated with Pauling's rules. 20 This picture implicitly assumes a strong coupling between the cationic and anionic sublattices. An alternative approach to the description of the structure was proposed by O'Keeffe and Hyde in the mi-80s: [START_REF] O'keeffe | An alternative approach to non-molecular crystal structures with emphasis on the arrangements of cations[END_REF] the oxide structures are described in terms of their cation packings and the coordination of the anions by these cations. In other words, instead of considering a structure as a regular anionic lattice into which cations have been inserted, the alternative is used, namely the insertion of anions into a regular cation array. [START_REF] O'keeffe | An alternative approach to non-molecular crystal structures with emphasis on the arrangements of cations[END_REF][START_REF] Ross | The Stability of Minerals[END_REF] Later, this idea was reintroduced through the "anions in metallic matrices" model, where a crystal structure of an inorganic compound can be described as a metallic matrix in which geometric and electronic structures govern the localization of the anions in the lattice. [START_REF] Vegas | [END_REF] The results obtained on SnO 2 under high-pressure can be interpreted in this framework: the Sn cation lattice ensures stability over a wide pressure range. However, since there is a weak coupling Defects and surface chemistry also influence structural stability. The different annealing temperatures of the as-prepared particles to obtain batch 1 (750 K) and batch 2 (1120 K) can induce certain modifications in the concentration of defects and the surface chemistry. However, it has been shown that defects have a destabilizing effect on the crystallographic structure leading to pressure-induced disordering (ultimately amorphization) and a decrease of the transition pressure. 9,18,42 Moreover, this effect is all the more marked as the particles are small. This is not what is observed in this study where the smallest particles show the greatest stability. This seems to indicate that the surface chemistry and the associated defects are not the essential parameters driving the structural stability (destabilization) of the nanoparticles.

The second stabilizing effect is linked to the solid solution. Indeed, while no phase transition has been reported using X-ray diffraction which mainly probes the cationic sub-lattice, a pressure-induced disordering has been observed in SnO 2 nanoparticles and SnO 2 bulk in using Raman spectroscopy. 12,32 This characterization technique is more sensitive to the anionic sub-lattice. Indeed, a pressure-induced disordering of the oxygen sublattice starting above 7 GPa in 3 nm particles was evidenced, with the propagation of the disorder from the shell to the core. 12 Such disordering ultimately induced an oxygen sub-lattice amorphization at P ∼ 19 GPa (Fig. 8). Likewise, it was recently shown that bulk SnO 2 also exhibits such a decoupling of cationic and anionic sublattices under pressure. 32 The oxygen sublattice in bulk SnO 2 indeed transforms to a disordered array while the cationic sub-lattice shows no transformation. Such behavior is similar to what is observed in nanoparticles. It has also been shown that a critical defect density (mainly oxygen vacancies) destabilizes the anionic sublattice without significantly affecting the metallic sublattice in SnO 2 . These observations indicate a general property associated to this material, i.e., a tendency to have a decoupling between cationic and anionic sublattices. 32 In the case of anatase TiO 2 , such effect of disordering is to a certain extent observed for nanoparticles smaller than 10 nm and exhibiting a critical defect density. 16,42 The difference with SnO 2 is that the two sublattices, anionic and cationic, undergo a disordering at the phase transition around 18-20 GPa, leading to a complete amorphization (Fig. 8). [15][16][17] Thus, in TiO 2 nanostructures (with anatase or rutile structure), a pressure-induced phase transitiontowards disordered baddeleyite or an amorphous state, depending on size and surface state of the particlesis always observed below 20 GPa without clear 1/R variation. 13,16,18 In the case of Ti 0.5 Sn 0.5 O 2 , the decoupling of the sublattices reported for SnO 2 is not observed even at high pressure despite the presence of defects, as evidenced by Raman spectroscopy. Therefore, the solid solution Ti 0.5 Sn 0.5 O 2 shows an extended stability at the nanoscale, both of its cationic and anionic sublattices, which is highly unusual and unexpected compared to the end members (Fig. 8).

This result can be interpreted in an original perspective of the description of phase transition which has been recently the deficit of oxygen does not play this role of buffer between cations leading to a higher stability of the bct or bcc structure as it is observed in the elemental Sn under pressure. Therefore, the smaller the nanoparticles, the higher the effect of surface and oxygen vacancies, and the bcc packing is stable as in the case of pure Sn. Therefore, the size-dependent stability of the rutile phase may be related to the stability of the cationic bcc lattice under pressure.

On the other hand, the absence of disordering of the oxygen sublattice in Ti 0.5 Sn 0.5 O 2 nanoparticles is the index of a stronger coupling between the sublattices unlike in SnO 2 nanoparticles. This effect most likely originates in the difference in cohesive energy (much lower in SnO 2 than in TiO 2 ) which reflects the energy of metal-ligand bonds. [START_REF] Medvedevva | [END_REF] This effect explains why the Raman spectrum of Ti 0.5 Sn 0.5 O 2 resembles that of TiO 2 (Fig. 2a). The main peaks of the rutile structure are only slightly affected unlike the case of SnO 2 which shows a completely different Raman spectrum affected by the defects. 34,35 This indicates a privileged coupling between the Ti and O sublattices.

Therefore, the extreme stability observed in ultrafine Ti 0.5 Sn 0.5 O 2 nanoparticles can be interpreted as the result of a synergistic effect between Sn and Ti atoms. Sn provides a sizedependency on the stability field and Ti induces a strong coupling between anionic and cationic sublattices, preventing the propagation of the disorder in the oxygen sublattice. In other words, each cation plays a respective role: Sn ensures the stability of the metallic sublattice with respect to pressure and Ti ensures the stability of the oxygen sublattice.

Conclusions

This work highlights the enhanced structural stability of the TiO 2 -SnO 2 system at the nanoscale. Under pressure, the ∼5 nm particles show no phase transformation up to ∼30 GPa. In the case of end-members, phase transitions are observed in the range 16-20 GPa. 3 nm SnO 2 nanoparticles exhibit a sublattice disordering starting from 7 GPa and ending at around 19 GPa, and the TiO 2 nanoparticles transform to a highpressure phase at P ∼ 16-18 GPa. For Ti 0.5 Sn 0.5 O 2 nanoparticles of ∼10 nm, a phase transition is observed between 20 and 25 GPa.

The size dependence of the transition pressure is similar to what is observed in pure SnO 2 . However, no pressure-induced disordering of the anionic sublattice was observed, contrary to what is observed in the SnO 2 nanoparticles. The experimental pieces of evidence are explained as follows: Ti cations introduce coupling between cationic and anionic sublattices (stronger metal-ligands bonds) and Sn cations improve the structural stability of the cationic sublattice. This leads to a synergistic effect to which the presence of the two atoms provides enhanced structural stability by counterbalancing their effects. This finding paves the way to the development of new nanomaterials that exhibit improved properties by selecting the favorable contribution of each component, similar to what is between cationic and anionic sublattices, the oxygen sublattice with an initial critical defect density evolves differently. Under pressure, the defect density increases until percolation and amorphization of the anionic network occur. 32 This effect has recently been confirmed by a combination of XRD and EXAFS. 47 The effect of decoupling between the two sublattices has been observed in pressure-induced phase transitions in complex structures of rare earth molybdates (RE(MoO 4 ) 3 with RE = Gd, Eu, Tb, Nd), for which two-step amorphization has been reported regardless of the form of the samples. The anionic sublattice begins to exhibit amorphous-like Raman, photoluminescence, or EXAFS signatures at a pressure lower than that of the cationic sublattice probed by X-ray diffraction. 48,49 Partial (sublattice) amorphization has also been discussed in Co(OH) 2 , where infrared spectroscopy reported a disordering of the O-H sublattice. 50 In Ga 2 O 3 , the β-to-α pressure-induced phase transition is accompanied by a strong broadening of the Raman signal while X-ray diffraction patterns do not indicate such a disordering in the cationic sublattice. 51 The rutile structure is particularly suited to such studies of a particular sublattice, since only oxygen atoms are involved in the Raman-active vibrational modes. 52 The rutile structure consists of a body centered tetragonal (bct) cationic array. This is the atomic arrangement found in the high-pressure γ-Sn allotrope 53,54 which is stable over a large range of pressure (from 10 to 31 GPa) showing only slight distortion to a bodycentered orthorhombic structure (bco) above 32 GPa and to a bcc lattice above 40 GPa. 54 In the case of Ti, the bct packing is metastable and is only observed under non-hydrostatic pressure [ref. 55 and refs therein]. However, the high-pressure hexagonal phase ω can be related to a distorted bcc structure. 55 According to the model proposed by Bain, bct is the intermediate structure in the bcc-to-fcc transformation. Pressure should favor the fcc due to its higher compacity in a hard sphere model, a situation found for the TiO 2 and SnO 2 oxides as the high-pressure phases observed either in TiO 2 (baddeleyite) or SnO 2 (distorted fluorite) correspond to distorted fcc cationic sublattices. 39,56 However, this situation in not found in Sn and Ti, where the bcc packing is stable under pressure. This stability of bcc packing under pressure has been discussed by Oganov et al. 57 in the case of calcium. This can be understood by the fact that bcc is the most compact structure because of shorter bond length i.e., the sphere diameter is not the same in the bcc or in the fcc packing. In some sense, this configuration corresponds to the packing of interpenetrating spheres (shorter distance if the diameter is fixed) for which the bcc has the highest density. [START_REF] Iglesias-Ham | Sphere packing with limited overlap[END_REF] Therefore, a possible explanation of the stability of the SnO 2 nanoparticles is the following: in bulk SnO 2 , the oxygen atoms prevent the interpenetration of cations (as in the case of pure Sn) which leads to a hard sphere behavior under pressure i.e., a transformation to a fcc arrangement of the cations (distorted fluorite). In the case of nanoparticles, the surface and

Fig. 1

 1 Fig. 1 (a) X-Ray diffractograms from batch 1 (treatment at 750 K during 4 h) and batch 2 (treatment at 1120 K during 2 h). (b) Typical TEM images obtained on batch 2. (c) Fast Fourier transform and indexation of the particle shown in (b).

Fig. 2

 2 Fig. 2 (a) Comparison of the Raman spectra for nanostructures of n-SnO 2 (3 nm nanoparticles), n-TiO 2 (nanorods with 8 nm diameter from ref. 14) and n-Ti 0.5 Sn 0.5 O 2 (5 nm nanoparticles) along with Raman spectra of bulk TiO 2 and SnO 2 (b-TiO 2 and b-SnO 2 ). All samples adopt a rutile-type structure. In b-TiO 2 , the B 2g mode is hardly detectable whereas in b-SnO 2 , the B 1g is not observed (see text). (b) Comparison between experimental Raman spectra and simulated vibrational density of states (vDOS) for Ti 0.5 Sn 0.5 O 2 .

Fig. 4

 4 Fig. 4 (a) Raman spectra of Ti 0.5 Sn 0.5 O 2 of 5 nm (batch 1) with increasing pressure. (b) Low-frequency range recorded during the same experiment.

Fig. 5

 5 Fig. 5 (a) Raman spectra showing the entire pressure cycle from ambient pressure to the highest pressure reached in this experiment (30.1 GPa) and followed by decompression to ambient pressure. (b) Low-frequency range recorded during the same experiment.

Fig. 6

 6 Fig. 6 (a) Raman spectra of Ti 0.5 Sn 0.5 O 2 of 10 nm (batch 2) with increasing pressure. (b) Low-frequency range recorded during the same experiment.

Fig. 7

 7 Fig. 7 (a) Raman spectra of Ti 0.5 Sn 0.5 O 2 of 10 nm (batch 2) with decreasing pressure. (b) Low-frequency range recorded during the same experiment.

Fig. 8

 8 Fig. 8 Structural representation along the c-axis of the initially rutile phase. The structure is identical at ambient pressure for SnO 2 , TiO 2 and Ti 0.5 Sn 0.5 O 2 differing only by the value of their cell parameters. Under pressure, in the presence of defects, TiO 2 nanoparticles undergo amorphization with the disordering of both cationic and anionic sublattices. In SnO 2 , the cationic sublattice provides structural stability, and only the oxygen sublattice experiences such a disordering because of a weaker Sn-O bonding. In Ti 0.5 Sn 0.5 O 2 , the Sn ensures the structural stability as in SnO 2 . The coupling between the cations and the anions is stronger due to the Ti-O bonds, leading to extreme structural stability.

Acknowledgements

The authors acknowledge the Consortium Lyon Saint-Etienne (CLYM) for the access to the electron microscope and the CECOMO and PLECE platforms at Université Lyon 1 for the use of the spectrometer and of the Diamond Anvil Cells, respectively.

Conflicts of interest

There are no conflicts to declare.


Philippe Balbiani

Coherence test on graphs constraints between hyperintervals

This paper proceeds to develop a model for representing and reasoning about time from the perspective of nonstandard analysis.

Introduction

The purpose of the temporal system about which we are going to speak is to allow the correct interpretation of expressions putting at stake various levels of granularity in a representation of time based on extended entities associated with several binary relations. It is important for at least two reasons. The first reason lies in the fact that a correct modelling of the temporal phenomena of the real world requires the consideration of various levels of granularity in the representation of time [START_REF] Euzenat | Time granularity In: Handbook of Temporal Reasoning in Artificial Intelligence[END_REF]. To model hybrid systems the behavior of which involves inevitably continuous and discrete constituents, the appeal to infinitesimal entities (hyperreals) associated with several binary relations (comparability, nearness, . . .) was often made [START_REF] Iwasaki | Modeling time in hybrid systems: how fast is "instantaneous"?[END_REF]. The second reason lies in the fact that a correct interpretation of what relates to time in natural languages requires the surpassing of the simple notion of temporal precedence between punctual entities as moments [START_REF] Bestougeff | Outils logiques pour le traitement du temps: de la linguistique à l'intelligence artificielle[END_REF]. To interpret statements corresponding to an activity which involves inevitably the existence of a beginning and an end, the appeal to extended entities (intervals) associated to several binary relations (strict precedence, overlapping, . . .) took an important development [START_REF] Van Beek | Approximation algorithms for temporal reasoning[END_REF].

Hyperreals and hyperintervals

In this paper, we shall be interested only in the set P H of all non-unlimited hyperreals. We can show that for all hyperreals a ∈ P H , there exists a real r a and an infinitesimal hyperreal a such that a = r a + a . r a is called standard part of a and a is called infinitesimal part of a. The writing of a as the addition of r a and a invites us to define on Let I H be the set of all hyperintervals. Among the relations which are possible between two hyperintervals, two are to be considered: the relation

= of direct meet such that [a, b] = [c, d] iff b = c and the relation -of shifted meet such that [a, b] -[c, d] iff b≺ inf c.
3 Hyperinterval algebra

Primitive relations

Let us notice that every hyperinterval

[b 1 , b 2 ] defines 9 zones in P H : Z 0 = {a ∈ P H : a≺ app b 1 }, Z 1-= {a ∈ P H : a≺ inf b 1 }, Z 1 = {b 1 }, Z 1+ = {a ∈ P H : b 1 ≺ inf a}, Z 2 = {a ∈ P H : b 1 ≺ app a and a≺ app b 2 }, Z 3-= {a ∈ P H : a≺ inf b 2 }, Z 3 = {b 2 }, Z 3+ = {a ∈ P H : b 2 ≺ inf a} and Z 4 = {a ∈ P H : b 2 ≺ app a}.
As regards the possible relative positions of two hyperintervals, we thus obtain 33 primitive relations, p, m -, m , m + , o, s -, s , s + , d, fi -, fi , fi + , eq -,-, eq -, , eq -,+ , eq ,-, eq , , eq ,+ , eq +,-, eq +, , eq +,+ , f -, f , f + , di, si -, si , si + , oi, mi -, mi , mi + and pi, among which one, eq , , is the identity. We shall denote the primitive relations by the letters A, B and C. In Fig. 1, for all primitive relations A, the value indicated at column "i A " is the unique element of {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} for which there exists hyperintervals

[a 1 , a 2 ], [b 1 , b 2 ] such that [a 1 , a 2 ] A [b 1 , b 2 ]
and a 1 belongs to the zone Z iA defined in P H by [b 1 , b 2 ] and the value indicated at column "j A " is the unique element of {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} for which there exists hyperintervals

[a 1 , a 2 ], [b 1 , b 2 ] such that [a 1 , a 2 ] A [b 1 , b 2 ]
and a 2 belongs to the zone Z jA defined in P H by [b 1 , b 2 ]. The column "Dimension of A" in Fig. 1 suggests to associate with every primitive relation A an element dim(A) in {0, 1, 2, 3, 4}. The column "Ordinal representation of A" in Fig. 1 suggests to associate with every primitive relation A a 2-tuple (i A , j A ) of elements in {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4}. Let dim(•) be the func-

A i A j A Dim. of A Ord. rep. of A p 0 0 4 (0, 0) m - 0 1- 3 (0, 1-) m 0 1 2 (0, 1 ) m + 0 1+ 3 (0, 1+) o 0 2 4 (0, 2) fi - 0 3- 3 (0, 3-) fi 0 3 2 (0, 3 ) fi + 0 3+ 3 (0, 3+) di 0 4 4 (0, 4) s - 1- 2 3 (1-, 2) eq -,- 1- 3- 2 (1-, 3-) eq -, 1- 3 1 (1-, 3 ) eq -,+ 1- 3+ 2 (1-, 3+) si - 1- 4 3 (1-, 4) s 1 2 2 (1 , 2) eq ,- 1 3- 1 (1 , 3-) eq , 1 3 0 (1 , 3 ) eq ,+ 1 3+ 1 (1 , 3+) si 1 4 2 (1 , 4) s + 1+ 2 3 (1+, 2) eq +,- 1+ 3- 2 (1+, 3-) eq +, 1+ 3 1 (1+, 3 ) eq +,+ 1+ 3+ 2 (1+, 3+) si + 1+ 4 3 (1+, 4) d 2 2 4 (2, 2) f - 2 3- 3 (2, 3-) f 2 3 2 (2, 3 ) f + 2 3+ 3 (2, 3+) oi 2 4 4 (2, 4) mi - 3- 4 3 (3-, 4) mi 3 4 2 (3 , 4) mi + 3+ 4
tion from {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} into {0, 1, 2}
defined by: for all n ∈ {0, 1, 2}, dim(2n) = 2 and for all n ∈ {1, 2}, dim((2n -1)-) = 1, dim((2n -1) ) = 0 and dim((2n -

1)+) = 1. Let ent(•) be the function from {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} into {0, 1, 2, 3, 4} defined by: for all n ∈ {0, 1, 2}, ent(2n) = 2n and for all n ∈ {1, 2}, ent((2n -1)-) = 2n -1, ent((2n -1) ) = 2n -1 and ent((2n -1)+) = 2n -1. Notice that for all primitive relations A, dim(A) = dim(i A ) + dim(j A ) and either i A is even or j A is even or ent(i A ) < ent(j A ).

Graphical representation

The columns "Dimension of A" and "Ordinal representation of A" suggests to represent every primitive relation A by Fig. • be the function from {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} into {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} defined by: for all n ∈ {0, 1, 2}, 2n = 2n and for all n ∈ {1, 2}, (2n -1)-= (2n -1)-, (2n -1) = (2n -1)and (2n -1)+ = (2n -1)and • be the function from {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} into {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} defined by: for all n ∈ {0, 1, 2}, 2n = 2n and for all n ∈ {1, 2}, (2n -1)-= (2n -1)+, (2n -1) = (2n -1)+ and (2n -1)+ = (2n -1)+. We shall say that an interval [i, j] of {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} is balanced iff either i and j are even or i = j or there exists n ∈ {1, 2} such that i = (2n -1)and j = (2n -1)+. For all intervals [i, j] of {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4}, we shall denote by [[i, j]] the greatest balanced interval of {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} contained in [i, j]. With every zone Z of the graphical representation of A, we associate a balanced interval int A (Z) of {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} as follows:

int A (Z 0 ) = [[0, i A ]], int A (Z 1-) = [[ i A , i A ]], int A (Z 1 ) = {i A }, int A (Z 1+ ) = [[i A , i A ]], int A (Z 2 ) = [[i A , j A ]], int A (Z 3-) = [[ j A , j A ]], int A (Z 3 ) = {j A }, int A (Z 3+ ) = [[j A , j A ]] and int A (Z 4 ) = [[j A , 4]].

Transposition and composition

, if i C ∈ int B (Z iA ) and j C ∈ int B (Z jA ) then C is in A • B.
A set of primitive relations is called a relation. We shall denote the relations by the letters R, S and T . For all relations R, S, we define:

0 = ∅, -R = {p, . . . , pi} \ R, R S = R ∪ S, id = {eq , }, R -1 = {A -1 : A ∈ R} and R • S = {A • B: A ∈ R and B ∈ S}.
Let R 33 be the set of all relations. With every relation R of R 33 , we associate a relation η(R) on I H as follows: a η(R) b iff there exists a primitive relation A such that A ∈ R and a A b. As the reader is asked to show, η(•) is an isomorphism from (R 33 , 0, -, , id, -1 , •) into a subalgebra of relations of the set of all relations on I H . Consequently,

Proposition 3 (R 33 , 0, -, , id, -1 , •) is a representable algebra of relations.
For all relations R, we define: dim(R) = max{dim(A): A ∈ R}.

Convex relations

p m p m + p o p fi - p fi p fi + p di p s - p eq -,-
p eq -, p eq -,+ p si - p s p eq ,- p eq , p eq ,+ p si p s + p eq +,- p eq +, I are such that the set of all primitive relations is a lattice. Obviously, for all relations R, R ⊆ I(R). Moreover, if R and S are relations such that R ⊆ S then I(R) ⊆ I(S). A relation R is said to be convex iff I(R) ⊆ R. For example, {eq ,+ , eq +,+ , f + , si , si + , oi} is convex.

p eq +,+ p si + p d p f - p f p f + p oi p mi -p mi p mi + p pi

We write A

I R if for all primitive relations C, if C is in R then A I C and R I B if for all primitive relations C, if C is in R then C I B. [A, B]
Proposition 4 Let R, S be relations. If R and S are convex then R S is convex. Proposition 5 Let R, S be relations. If R and S are convex then R • S is convex. Obviously, I(R) is convex. Moreover, Proposition 6 Let R, S be relations. 1. I(R -1 ) = I(R) -1 .

I(R • S) ⊆ I(R) • I(S).

Preconvex relations

Let C be the least order on {0, 1-, 1 , 1+, 2, 3-, 3 , 3+, 4} such that 0 C 1-, 1-C 1 , 1+ C 1 , 2 C 1+, 2 C 3-, 3-C 3 , 3+ C 3 and 4 C 3+.
The order C between primitive relations represented in Fig. 4 is definable as follows:

A C B iff i A C i B and j A C j B . C (A) will designate the p 0 p 1- p 1 p 1+ p 2 p 3- p 3 p 3+ p 4 p 0 p 1- p 1 p 1+ p 2 p 3- p 3 p 3+ p 4 p p p m - p m p m + p o p fi - p fi p fi + p di p s - p eq -,-
p eq -, p eq -,+ p si - p s p eq ,- p eq , p eq ,+ p si p s + p eq +,- p eq +, p eq +,+ set of all primitive relations B such that A C B. For all relations R, let CL(R) = { C (A): A ∈ R}. Obviously, for all relations R, R ⊆ CL(R). Moreover, if R and S are relations such that R ⊆ S then CL(R) ⊆ CL(S). We will say that a relation R is weakly preconvex if I(R) ⊆ CL(R). For example, {eq +,+ , f + , si , oi} is weakly preconvex.

p si + p d p f - p f p f + p oi p mi -p mi p mi + p pi
Proposition 7 Let R be a relation. dim(CL(R) \ R) < dim(R). Proposition 8 Let R be a relation. If R is weakly preconvex then dim(I(R) \ R) < dim(R).
We will say that a relation R is strongly preconvex if for all relations S, if S is convex then R S is weakly preconvex. For example, {eq +,+ , f + , si , si + , oi} is strongly preconvex. Obviously, if R is strongly preconvex then R is weakly preconvex. Moreover, Proposition 9 Let R, S be relations. If R is strongly preconvex and S is convex then R S is strongly preconvex.

Constraint satisfaction problems

The 3 previous sections naturally suggest us to consider the problem of the coherence of graphs constraints between hyperintervals. A graph G = (n, C) of constraints between hyperintervals consists of a positive integer n and a function C(•, •) from {1, . . . , n} × {1, . . . , n} into R 33 . We will say that G is nonempty if for all i, j ∈ {1, . . . , n}, Card(C(i, j)) ≥ 1. G is said to be coherent if there exists hyperintervals a 1 , . . . , a n such that for all i, j ∈ {1, . . . , n}, there exists a primitive relation A such that A ∈ C(i, j) and a i A a j . The problem of the coherence of graphs constraints between hyperintervals is the following: given a graph G = (n, C) of constraints between hyperintervals, determine whether G is coherent or not. We will say that G is atomic if for all i, j ∈ {1, . . . , n}, Card(C(i, j)) ≤ 1.

Proposition 10 The problem of the coherence of atomic graphs constraints between hyperintervals is in P .

Hence, The problem of the coherence of graphs constraints between hyperintervals is in NP . The trouble is that The problem of the coherence of graphs constraints between hyperintervals is NP -hard. The 2 following notions (convexity and path-coherence) have been considered together within the context of interval algebra [START_REF] Van Beek | Approximation algorithms for temporal reasoning[END_REF]. G is said to be convex if for all i, j ∈ {1, . . . , n}, C(i, j) is convex. We will say that G is path-coherent if for all i, j, k ∈ {1, . . . , n}, C(i, j) ⊆ C(i, k) • C(k, j). Let G = (n, C) be a convex graph constraints between hyperintervals. It can be proved that if G is nonempty and path-coherent then G is coherent. Thus, Proposition 11 The problem of the coherence of convex graphs constraints between hyperintervals is in P .

The 2 following notions (strong preconvexity and weak path-coherence) have been considered together within the context of rectangle algebra [START_REF] Balbiani | A new tractable subclass of the rectangle algebra[END_REF]. G is said to be strongly preconvex if for all i, j ∈ {1, . . . , n}, C(i, j) is strongly preconvex. We will say that G is weakly path-coherent if for all i, j, k ∈ {1, . . . , n}, C(i, j) ⊆ I(C(i, k) • C(k, j)). Let G = (n, C) be a strongly preconvex graph constraints between hyperintervals. It can be proved that if G is nonempty and weakly path-coherent then G is coherent. Thus, Proposition 12 The problem of the coherence of strongly preconvex graphs constraints between hyperintervals is in P .

Conclusion

Much remains to be done. At the level of the problem of the coherence of graphs constraints between hyperintervals, for example, we do not know if it is possible to consider sets of relations greater than the set of all strongly preconvex relations and for which the problem of the coherence of graphs constraints between hyperintervals remains feasible in polynomial time. In other respect, there is the question of the development for the hyperreals of a qualitative approach à la Ligozat [START_REF] Ligozat | On generalized interval calculi[END_REF]. For the moment, we know nothing of the feasibility of the problem of the coherence of graphs constraints between generalized hyperintervals that such a development would arouse.

P

  H the relations ≺ inf and ≺ app by: a≺ inf b iff r a = r b and a ≺ b and a≺ app b iff r a ≺r b . Hyperreals a, b ∈ P H such that a≺ app b define an hyperinterval, denoted by [a, b].

Figure 1 .

 1 Figure 1. Primitive relations, zones, dimensions and ordinal representations.

Figure 2 .

 2 Figure 2. Graphical representation.

LetProposition 1 Proposition 2

 12 A be a primitive relation. As the reader is asked to show, there exists a unique primitive relation, denoted by A -1 and called the transposition of A, such that there exists hyperintervals a, b such that a A -1 b and b A a. Moreover, there exists hyperintervals a, b such that a A -1 b and bA a iff 1 ∈ int A (Z i A -1 ) and 3 ∈ int A (Z j A -1 ). Consequently, Let A be a primitive relation. The transposition A -1 of A is the unique primitive relation such that 1 ∈ int A (Z i A -1 ) and 3 ∈ int A (Z j A -1 ).Let A, B be primitive relations. As the reader is asked to show, there exists a least set of primitive relations, denoted by A • B and called the composition of A and B, such that for all primitive relations C, if there exists hyperintervals a, b, c such that a C b, a A c and c B b then C is in A • B. Moreover, there exists hyperintervals a, b, c such that a C b, a A c and c B b iff i C ∈ int B (Z iA ) and j C ∈ int B (Z jA ). Consequently, Let A, B be primitive relations. The composition A•B of A and B is the least set of primitive relations such that for all primitive relations C

1 -, 1 , 3 I

 113 1+, 2, 3-, 3 , 3+, 4} such that 0 I 1-, 1-I 1 , 1 I 1+, 1+ I 2, 2 I 3-, 3-I 3 , 3+ and 3+ I 4. The order between primitive relations represented in Fig.3is definable as follows: A I B iff i A I i B and j A I j B . The properties of the order p

Figure 3 .

 3 Figure 3. Lattice of primitive relations.

  will designate the set of all primitive relations C such that A I C and C I B. For all relations R, let I(R) = {[A, B]: A I R and R I B}.

Figure 4 .

 4 Figure 4. Closure of primitive relations.


domain: phys chim

text:
Sumit Ranjan Maity

Monica Ceretti

Lukas Keller

Ju¨rg Schefer

Bertrand Roessli

Uwe Stuhr

Christof Niedermayer

Werner Paulus

Evidence of correlated incommensurate structural and magnetic order in highly oxygen-doped layered nickelate Nd2NiO4.23

Nd2NiO4+δ is a non-stoichiometric oxide, crystallizing in the Ruddlesden-Popper type framework, exhibiting a wide range of oxygen non-stoichiometry (δ) with a complex structural phase diagram. Excess oxygen atoms insert holes, directly influencing the Ni valence state and associated electronic structure. We report here the existence of a correlated incommensurate structural and antiferromagnetic order below TN ≈ 150 K in highly oxygen-doped Nd2NiO4+δ with δ ≈ 0.23. The crystal and magnetic structures are investigated by polarized and unpolarized single crystal neutron diffraction studies together with synchrotron X-ray powder diffraction and macroscopic magnetization measurements. Due to the long-range ordering of excess oxygen atoms, the real structure of the compound is incommensurately modulated represented by Qn = ±0.813a * ±0.519b * wave vectors. The antiferromagnetic order is characterized by the presence of incommensurate Bragg peaks of type (h ± ε, 0, l/2) with h and l being odd integer numbers and the magnetic incommensurability ε ≈ 0.36. The inplane magnetic correlation length is found to be ξab = 184(1) ˚A, while significantly reduced along the caxis to ξc = 39(2) ˚A, indicating the quasi-two-dimensional (2D) nature of the antiferromagnetic correlations. Our study indicates that the excess oxygen atoms nevertheless strongly enhance the magnetic correlations between the NiO2-planes, resulting in a doubling of the magnetic unit cell along the c-axis. Contrary to Nd2NiO4.1, where the incommensurate periodicity of oxygen and magnetic ordering was found to be identical, the structural and electronic ordering in heavily oxygen-doped Nd2NiO4.23 show two independent modulation vectors. Our study thus unravels a strong correlation between structural and electronic orderings in highly oxygen-doped Ln2NiO4+δ-system.

I.

INTRODUCTION

The influence of oxygen non-stoichiometry (δ) on structural and physical properties of Ruddlesden-Popper (RP) Ln2NiO4+δ (Ln: La, Pr, Nd) compounds has been widely investigated since the discovery of superconductivity in iso-structural cuprates [START_REF] Tranquada | Oxygen intercalation, stage ordering, and phase separation in La2NiO4+δ with 0.05 ≤ δ ≤ 0.11[END_REF][START_REF] Poirot | Magnetic Properties of La2NiO4.16 and La12-xPrxNiO4+δ[END_REF][START_REF] Tranquada | Ordering of holes and spins in La2NiO4.125 and La1.8Sr0.2NiO4[END_REF][START_REF] Batlle | Weak ferromagnetism and magnetic interactions in La2NiO4[END_REF][START_REF] Demourgues | Transport and Magnetic Properties of La2NiO4+δ (0 ≤ δ ≤ 0.25)[END_REF][START_REF] Wochner | Neutron-diffraction study of stripe order in La2NiO4 with δ = 2/15[END_REF][START_REF] Nakajima | Neutron Scattering Study of Oxygen Ordering and Magnetic Correlations in La2NiO4[END_REF][START_REF] Paulus | Neutron diffraction studies of stoichiometric and oxygen intercalated La2NiO4 single crystals[END_REF][START_REF] Bassat | Anisotropic Oxygen Diffusion Properties in Pr2NiO4+δ and Nd2NiO4+δ Single Crystals[END_REF][START_REF] Ceretti | Low temperature oxygen diffusion mechanisms in Nd2NiO4+δ and Pr2NiO4+δ via large anharmonic displacements, explored by single crystal neutron diffraction[END_REF][START_REF] Perrichon | Lattice Dynamics Modified by Excess Oxygen in Nd2NiO4+δ: Triggering Low-Temperature Oxygen Diffusion[END_REF][START_REF] Nikonov | Two-dimensional quantum ferromagnet, magnetic chains, and oxygen ordering in La2NiO4.125 single crystals[END_REF]. The crystal structure of these compounds, as depicted in Fig. 1(a), can be described as stacking of LnNiO3-type perovskite layers interspersed with rock-salt type double LnO layers. The excess oxygen atoms are incorporated between two adjacent LnO layers in the tetrahedral interstitial sites coordinated by four apical oxygen atoms [START_REF] Jorgensen | Structure of the interstitial oxygen defect in La2NiO4+δ[END_REF]. The excess oxygen content δ can be tuned specifically in the range of 0≤ δ ≤0.25 [START_REF] Bhavaraju | Electrochemical Intercalation of Oxygen in Nd2NiO4+x (0 ≤ x ≤ 0.18) at 298 K[END_REF].

The doped oxygen atoms inject holes into the system by extracting electrons from the NiO2 planes, thereby modifying the valence states of Ni 2+ cations. Hole-doping could also be achieved by partially replacing the rare-earth cations with alkaline earth metals such as Sr 2+ , leading to Ln2-xSrxNiO4+δ. On cooling, these holes segregate into narrow channels that act as antiphase domain walls separating antiferromagnetic (AFM) regions of the Ni 2+ spins, known as a charge-stripe ordered state in which the spin order is represented by a modulation vector Qso=(QAF ± ε, 0, 0) where QAF=(1, 0, 0) is the commensurate antiferromagnetic wave vector observed for the undoped compound while the charge order is characterized by a modulation vector Qco=(QAF ± 2ε, 0, 0) with respect to the pseudo-tetragonal chemical unit cell [START_REF] Wochner | Neutron-diffraction study of stripe order in La2NiO4 with δ = 2/15[END_REF][START_REF] Tranquada | Stripe correlations of spins and holes in cuprates and nickelates[END_REF][START_REF] Tranquada | Evidence for stripe correlations of spins and holes in copper oxide superconductors[END_REF]. The magnetic incommensurability ε strongly depends on the hole-concentration (nh = x for Sr-doped systems and 2δ for oxygen-doped systems), and it was shown that ε changes linearly with nh for x < 1/2 [START_REF] Yoshizawa | Stripe order at low temperatures in La2-xSrxNiO4 with 0.289 ≲ δ ≲ 0.5[END_REF].

The hole-doping via oxygen insertion is not at all similar to Sr-doping, as in the oxygen-doped systems, in addition to electronic ordering at lower temperatures, the ordering of excess oxygen atoms also occurs at very moderate temperatures (T ≤ 800 K). For example, in La2NiO4+δ phases, a one-dimensional (1D) stacking of excess oxygen atoms was found for δ ≤ 0.11 in which intercalated layers of oxygen atoms spaced periodically along the c-axis and a three-dimensional (3D) order is observed for δ > 0.145, while both orders coexist for 0.11 ≤ δ ≤ 0.145 [START_REF] Zhang | Excess oxygen ordering in the La2NiO4+δ system studied by lowfrequency internal friction[END_REF]. The ordering of excess oxygen atoms leads to ordered deformation of NiO2 planes which provides a modulated potential that can pin the charge stripes to the lattice [START_REF] Zhang | Excess oxygen ordering in the La2NiO4+δ system studied by lowfrequency internal friction[END_REF]. Therefore, the excess oxygen atoms and their ordering state both play crucial roles in the nature of charge and spin orders. Single crystal neutron diffraction measurements on La2NiO4+δ compounds illustrated that the nature of antiferromagnetic order changes from commensurate to stripe at δ ≈ 0.125 and the charge order also appears at the same hole concentration [START_REF] Tranquada | Oxygen intercalation, stage ordering, and phase separation in La2NiO4+δ with 0.05 ≤ δ ≤ 0.11[END_REF]. A relationship between the modulation vectors of the charge order and the interstitial order was reported for La2NiO4.125 [START_REF] Wochner | Neutron-diffraction study of stripe order in La2NiO4 with δ = 2/15[END_REF]. Moreover, a correlation between structural and magnetic ordering was recently evidenced for moderately oxygen-doped Nd2NiO4.10 in which identical wave vectors were determined for the ordering of spins and excess oxygen atoms [START_REF] Maity | Interdependent scaling of long-range oxygen and magnetic ordering in nonstoichiometric Nd2NiO4.10[END_REF].

While the relationship between structural and electronic order has been comprehensively investigated for low and moderately oxygen-doped systems, it is not much explored for the highly oxygen-doped compounds. Moreover, no evidence of magnetic ordering was found in oxidized Ln2NiO4+δ compounds with δ ≥ 0.15 down to 1.5 K [START_REF] Rodr´ıguez-Carvajal | Structural Phase Transitions and Three-Dimensional Magnetic Ordering in the Nd2NiO4 Oxide[END_REF][START_REF] Rodriguez-Carvajal | Neutron diffraction study on structural and magnetic properties of La2NiO4[END_REF]. In this work, we investigate the ordering of spins in highly oxygen doped Nd2NiO4.23 and its correlation The details of indexation process can be found elsewhere [START_REF] Maity | Large-scale oxygen order phase transitions and fast ordering kinetics at moderate temperatures in Nd2NiO4+δ electrodes[END_REF]. (d) Refined anisotropic thermal ellipsoids of the apical (Oap) and equatorial oxygen (Oeq) atoms demonstrating strong structural disorder along the ab-plane and the caxis, respectively. The refined bond lengths shown here are obtained from an average structure refinement while the real structure is strongly distorted, corresponding to a tilt of ca. 25 • for the NiO6 octahedra and a shift of about 2 ˚A of apical oxygen atoms towards [110] with Oint-Oap distances of around 2.8 ˚A instead of 2.13 ˚A for the non-distorted tetrahedron [START_REF] Ceretti | Low temperature oxygen diffusion mechanisms in Nd2NiO4+δ and Pr2NiO4+δ via large anharmonic displacements, explored by single crystal neutron diffraction[END_REF][START_REF] Maity | Large-scale oxygen order phase transitions and fast ordering kinetics at moderate temperatures in Nd2NiO4+δ electrodes[END_REF]. (e) Temperature dependent DC magnetic susceptibility curves of the Nd2NiO4.23 single crystal recorded with an applied magnetic field of 1 T parallel to the c-axis and the ab-plane. The inset represents a first-order temperature derivative of the inverse magnetic susceptibility data. (f) Isothermal magnetization curves measured at 2 K and 300 K with magnetic fields applied parallel to the c-axis and the ab-plane. (g) First-order field derivative of the magnetization data shown in (f).

( e ) ( f ) ( g ) ( a ) ( b ) ( c ) ( d )
with the excess oxygen atoms by employing neutron and synchrotron Xray diffraction measurements.

II. EXPERIMENTAL METHODS

The details of powder sample preparation and single crystal growth of Nd2NiO4.23 is reported elsewhere [START_REF] Maity | Structural disorder and magnetic correlations driven by oxygen doping in Nd2NiO4+δ (δ ∼ 0.11)[END_REF][START_REF] Wahyudi | Growth of high quality single crystals of strontium doped (Nd,Pr)-nickelates, Nd2-xSrxNiO4+δ and Pr2-xSrxNiO4+δ[END_REF]. A single crystal of size approximately 2×2×2 mm 3 was used for all experiments. Synchrotron X-ray powder diffraction (SXRPD) measurements were performed at different temperatures between 5-300 K at the Material Sciences (MS) beamline X04SA (λ=0.564(1) ˚A) at the Swiss light source (SLS) of the Paul Scherrer Institute (PSI), Switzerland [START_REF] Willmott | The Materials Science beamline upgrade at the Swiss Light Source[END_REF]. Temperature-dependent DC magnetic susceptibility (χ) measurements on the single crystal were performed using a SQUID magnetometer in zero-field-cooled (ZFC) configuration by applying 1 T magnetic field parallel (χab; Hap ∥ ab) and perpendicular (χc; Hap ⊥ ab) to the NiO2 plane. Direction-dependent isothermal magnetization measurements were performed as a function of applied magnetic fields with decreasing fields from 9 T using a Physical Property Measurement System (PPMS) manufactured by Quantum Design.

Neutron diffraction reciprocal plane maps were performed at different temperatures on DMC at SINQ, PSI (λ=2.458(1) ˚A) [START_REF] Schefer | A versatile double-axis multicounter neutron powder diffractometer[END_REF][START_REF]Cold Neutron Powder Diffractometer DMC[END_REF]. Selected constant energy Q scans were carried out at various temperatures on the tripleaxis spectrometer EIGER at SINQ, PSI [START_REF] Stuhr | The thermal triple-axis-spectrometer EIGER at the continuous spallation source SINQ[END_REF][START_REF] Eiger | Thermal neutron triple-axis-spectrometer[END_REF]. Effective horizontal collimations of 40'-40'-40'-40' from the reactor to detector were used together with a pyrolytic graphite filter to eliminate neutrons at shorter harmonic wavelengths. Neutrons of energy 14.68 meV were selected. Polarized neutron diffraction measurements were performed on the triple-axis spectrometer TASP at SINQ, PSI (λ=3.14 ˚A) equipped with a MUPAD device [START_REF] Janoschek | Spherical neutron polarimetry with MuPAD[END_REF][START_REF]TASP: Cold Neutron Triple-axis spectrometer[END_REF]. Field-dependent magnetic measurements were carried out on the cold neutron tripleaxis spectrometer RITA-2 at SINQ, PSI using a 7 T horizontal magnet [START_REF] Lefmann | Added flexibility in triple axis spectrometers: The two RITAs at Risø[END_REF]. At all spectrometers, the neutrons were monochromatized and analyzed by pyrolytic graphite crystals set for the (002) reflection.

III. RESULTS AND DISCUSSIONS

A.

Modulated Crystal Structure

Room temperature synchrotron X-ray powder diffraction (SXRPD) data, as shown in Fig. 1(c), reveals a large number of superstructure reflections in addition to the sharp structural Bragg peaks, indicating the longrange ordering of excess oxygen atoms. The positions and intensities of the sharp Bragg peaks are explained by the low temperature monoclinic (LTM) structure with F112/m space group while most of the superstructure reflections are characterized by modulation vectors Qn = ±0.813a * ±0.519b * , as illustrated in the inset of Fig. 1(c). The details of the refinement and indexation procedure can be found elsewhere [START_REF] Maity | Large-scale oxygen order phase transitions and fast ordering kinetics at moderate temperatures in Nd2NiO4+δ electrodes[END_REF]. No structural transition is observed in the temperature range of 5-300 K and all the superstructures remain fixed to their positions with no significant intensity changes. The refined lattice parameters of the average F-centered unit cell are determined to be a = 5.3908(1) ˚A, b = 5.4510(1) ˚A, c = 12.3695(2) ˚A, α = β = 90 • and γ = 90.0764(1) • at 300 K. Refined anisotropic thermal ellipsoids of apical and equatorial oxygen atoms are shown in Fig. 1(d), showing large amplitudes along the ab-plane and the c-axis, respectively. These displacement amplitudes are induced by the excess oxygen atoms and are found to be essential for oxygen mobility close to room temperature within the framework of a phonon-assisted oxygen diffusion as recently demonstrated for oxygen non-stoichiometric oxides with Ruddlesden-Popper structures [START_REF] Ceretti | Low temperature oxygen diffusion mechanisms in Nd2NiO4+δ and Pr2NiO4+δ via large anharmonic displacements, explored by single crystal neutron diffraction[END_REF][START_REF] Maity | Large-scale oxygen order phase transitions and fast ordering kinetics at moderate temperatures in Nd2NiO4+δ electrodes[END_REF][START_REF] Maity | Structural disorder and magnetic correlations driven by oxygen doping in Nd2NiO4+δ (δ ∼ 0.11)[END_REF][START_REF] Ceretti | Nd/Pr)2NiO4+δ: Reaction Intermediates and Redox Behavior Explored by in Situ Neutron Powder Diffraction during Electrochemical Oxygen Intercalation[END_REF].

B. Magnetic Susceptibility

DC magnetic susceptibility curves are shown in Fig. 1(e). No apparent anomaly is observed in χ as a function of temperature since it is strongly dominated by the large, weakly coupled Nd moments while the effects of Ni moments are relatively small. However, three magnetic transitions could easily be distinguished from the temperature derivative of inverse magnetic susceptibility data as shown in the inset of Fig. 1(e). The first slope change at TN ≈ 150 K is related to the magnetic transition of Ni sub-lattice while the anomaly at 16 K is related to the reorientation of Ni magnetic moments along the crystallographic c-axis as we discuss in the following with neutron diffraction measurements. The anomaly at 8 K corresponds to the onset of magnetic order in the Nd sublattice as reported previously for other Nd2NiO4+δ compounds [START_REF] Rodr´ıguez-Carvajal | Structural Phase Transitions and Three-Dimensional Magnetic Ordering in the Nd2NiO4 Oxide[END_REF][START_REF] Maity | Structural disorder and magnetic correlations driven by oxygen doping in Nd2NiO4+δ (δ ∼ 0.11)[END_REF]. The magnetic susceptibility data were fitted using the CurieWeiss law given by , where C is the Curie constant and Θ is the Curie-Weiss temperature. Since, there is a strong contribution of Nd 3+ crystal field levels below 150 K, the fits were restricted in the temperature range of 150-330 K. The fitting parameters are listed in Table I. The negative Curie-Weiss temperature Θ indicates dominant antiferromagnetic interactions among and inbetween the Ni and Nd magnetic moments at low temperature.

Field-dependent isothermal magnetization curves are shown in Fig. 1(f) while differential magnetization curves are presented in Fig. 1(g). At 300 K, the magnetization curve is linear with the field, indicating a paramagnetic response of the system. However, these magnetization curves at lower temperatures below 10 K, in both directions, are non-linear especially at higher magnetic fields (> 3 T). We note that the magnetization curves, in both directions, are far from the magnetic saturation even at 2 K with 9 T magnetic field. A sharp fieldinduced magnetic transition is observed at Hap ≈ 3.5 T ∥ c-axis, illustrating the rotation of magnetic moments along this direction under the magnetic field. This transition is clearly visible in the differential magnetization curve as shown in Fig. 1(g), and only observed at T ≤ 8 K, demonstrating its strong correlation to the Nd magnetic moments.

C.

Magnetic structure Reconstructed neutron diffraction (h0l) reciprocal plane map obtained at 2 K is displayed in Fig. 2(a). Reciprocal h and k directions were arbitrarily assigned since the crystal is twinned on (hk0) plane. Thus, our following discussion is equally valid for the (0kl) reciprocal plane. Along with sharp structural Bragg peaks, diffuse type incommensurate magnetic satellites at (h ± ε, 0, l/2) positions are observed with h and l being odd integer numbers where ε is the observed magnetic incommensurability. Representative one-dimensional reciprocal h and l cuts through the (0.64, 0, 1.5) satellite are illustrated in Fig. 2(b) and (c), revealing the magnetic incommensurability ε ≈ to be 0.36 r.l.u. with centering at l=n/2 where n is an odd integer number. No temperature dependence in the modulus of ε is observed in our investigated temperature range. The integrated intensity of the (0.64, 0, 1.5) satellite is shown as a function of temperature in Fig. 2(d), demonstrating the appearance of an antiferromagnetic order below 150 K, in accordance to our DC magnetic susceptibility data. The magnetic correlation lengths are obtained from the fitting of reciprocal h and l scans with Lorentzian peak shapes convoluted with Gaussian-type instrumental resolution function, which was estimated from the fitting of sharp structural Bragg peaks. The magnetic correlation lengths from the Lorentzian peak widths are found to be 184 [START_REF] Perrichon | Lattice Dynamics Modified by Excess Oxygen in Nd2NiO4+δ: Triggering Low-Temperature Oxygen Diffusion[END_REF] ˚A and 39(2) ˚A in the ab-plane (ξab) and along the c-axis (ξc) at 2 K, respectively, suggesting the quasi-2D nature of antiferromagnetic correlations. The observed correlation length ξc is strongly increased compared to 8.4 ˚A that is found in Sr-doped Pr2-xSrxNiO4 (x ≈ 0.5) [34], demonstrating the role of ordered excess oxygen atoms in the interstitial sites on the inter-plane magnetic correlation. The temperature dependence of peak widths is presented in the Fig. S1 of the Supplemental Material for the magnetic satellite peak at (0.64, 0, 1.5). No pronounced change in the peak width is observed in the ab-plane as a function of temperature while a clear increase is observed along the c-axis i.e. perpendicular to the NiO2-planes, indicating a substantial decrease in the inter-plane magnetic correlations on approaching the magnetic transition temperature. We note that the observed magnetic phase in this study seemingly differs from a charge-stripe ordered state that is usually observed in the Sr-doped and moderately oxygen doped nickelates [START_REF] Tranquada | Evidence for stripe correlations of spins and holes in copper oxide superconductors[END_REF][START_REF] Freeman | Spin reorientation transition in the incommensurate stripe-ordered phase of La3/2Sr1/2NiO4[END_REF][START_REF] Freeman | Stripe order and magnetic transitions in La2-xSrxNiO4[END_REF]. This is justified not only by the absence of magnetic reflections at l=integer numbers but also by the observed ε value which is determined to be substantially lower than the expected value of nh=2δ ≈ 0.46 for a chargestripe ordered state as previously demonstrated for La2NiO4+δ compounds with similar hole concentration.

Reciprocal plane maps

Spherical neutron polarimetry

The diffuse incommensurate reflections observed in our plane maps were previously reported for a highly oxygendoped La2NiO4.152 compound [START_REF] Yamada | Incommensurate magnetic correlations in La2NiO4.125, Magnetic phase diagram and twodimensional hole-order[END_REF]. However, the origin of these reflections was unclear and speculated to be from an unknown long-range magnetic order. In this study, we have employed the spherical neutron polarimetry technique to further explore the origin of these diffuse incommensurate peaks and to estimate the orientation of magnetic moments. This process can be described by a polarization matrix Pαβ, which consists of 18 different scattering intensities in the spin-flip σ(α,-β) and non-spin-TABLE II. Measured and calculated polarization matrices for Q1 = (0.64, 0, 1.5) and Q2 = (0.64, 0, 4.5) magnetic reflections at 2 K and 20 K. G1, G2, and G3 magnetic models were used for matrix elements calculations with MuFit program for the 2 K data only. Since integrated intensities were not measured at 20 K, we don't have any magnetic model and calculated matrix elements for this temperature. Matrices were recorded with MUPAD on TASP at SINQ, PSI.

Measured Calculated G1 G2 G3

flip σ(α,β) channels in three dimensions [START_REF] Janoschek | Spherical neutron polarimetry with MuPAD[END_REF][START_REF] Blume | Polarization effects in the magnetic elastic scattering of slow neutrons[END_REF]. The initial and final spin directions are defined by α and β such that, P( where α and β: x,y,z

The set of polarization axes is defined as follows: the x direction is parallel to Q, the z-direction is perpendicular to the scattering plane and the y-direction completes the right-handed coordinate system. During our measurement, the crystal was mounted on the (h0l) scattering plane with the z-axis parallel to the crystallographic baxis of the crystal. Typical scan profiles of the (0.64, 0, 1.5) reflection are plotted in Fig. 2(e) for non spinflip (xx) and spin-flip (x-x) configurations. Nearly ten times larger intensity is detected in the spin-flip channel compared to the non-spin-flip channel, demonstrating the magnetic origin of the diffuse satellites. Complete polarization matrices were recorded at 2 K for two magnetic satellites at Q1=(0.64, 0, 1.5) and Q2=(0.64, 0, 4.5). Polarization matrices were also measured for two nuclear reflections at (2, 0, 0) and (0, 0, 2) to estimate the polarization efficiency which was determined to be at 96%. Furthermore, a complete polarization matrix was recorded for a background point at Q3=(0.68, 0, 1.4) at 2 K to eliminate the contributions of systematic errors.

Group-theory analysis

In this work, we attempt a group theoretical approach to obtain a magnetic model by using the integrated intensity and polarimetry data for a set of magnetic Bragg peaks. The magnetic propagation vector for the compound can be taken as k = (0.64, 0, 0.5). The average structural symmetry of the paramagnetic phase can be assumed Fmmm space group by discarding the subtle monoclinic structural distortion. The program BasIrreps was used to calculate the possible irreducible magnetic representations (IRs) valid with the propagation vector k. The magnetic representation at the Ni and Nd sites can be decomposed into a direct sum of IRs as Γmag = 1Γ(1) + 2Γ [START_REF] Poirot | Magnetic Properties of La2NiO4.16 and La12-xPrxNiO4+δ[END_REF]. The Nd site is split into two orbits related by a center of inversion. All calculated IRs are onedimensional. The basis vectors III. Least-squares refinements results of the integrated intensity data obtained for three different magnetic models at 2 K.

RF(%) RF2(%)

Ni-site at (0, 0, 0) Nd-site at (0, 0, z) ψ for different IRs are calculated using the projection operator technique. The spin distribution of the j-th atom can be expressed as the Fourier transform of the linear combination of basis vectors, such that for a single magnetic propagation vector sj = P n Cnψnexp(ik.t) + c.c where the coefficients Cn can be complex, in general. Magnetic moments, for the Ni and Nd sites at (000) and (00z) sites respectively, are obtained from the basis function calculated for the Γ(1) and Γ(2) resolved along the crystallographic axes as (0, v, 0) and (u, 0, w), respectively. Magnetic structure refinements allows the determination of the free parameters u, v and w experimentally. That means the magnetic moments are either pointing along the b-axis (i.e. in Γ(1), hereafter denoted as the G1 model) or lying on the ac-plane (i.e. in Γ(2), hereafter denoted as the G2 model). To check which model represents our data best, least-square refinements of the integrated intensity data were carried out. Integrated intensities of 18 magnetic reflections were collected with typical ω-scans with the crystal oriented on the (h0l) reciprocal plane. In addition, the integrated intensities of six structural Bragg reflections were collected to derive the scale factor for the magnetic structure analysis.

µx
Surprisingly, it was found with the least-square refinements that both G1 and G2 models provide a similar quality of fit to the observed integrated intensity data (See Fig. S2 in the Supplemental Material). Therefore, we looked into polarization matrices obtained for the magnetic peaks to find the correct model. The recorded matrix elements for two magnetic satellites are presented in Table II after background corrections. We note that at 2 K, for these two satellites, P(x, x) ̸ = P(y, y) ̸ = P(z, z) and both P(x, x) and P(y, y) are negative while P(z, z) is positive and all the other elements are zero within the statistics of the measurements except the mixed term P(y, z) for (0.64, 0, 1.5). These values strongly demonstrate pure magnetic character of both reflections. However, the P(y, z) component for (0.64, 0, 1.5) arises due to the interference effect between magnetic moments along y and z-directions [START_REF] Janoschek | Spherical neutron polarimetry with MuPAD[END_REF].

To determine which magnetic model is consistent with the observed polarization matrices, we have calculated the polarization matrices for these two magnetic satellites using both G1 and G2 magnetic models with the program MuFit assuming a single-domain sample [START_REF] Poole | Analysis of Neutron Polarimetry data using MuFit[END_REF]. Calculated polarization matrices for different models are also presented in Table II. It turns out that the G1 magnetic model is qualitatively in accordance with the observed polarimetry data. In contrast, the G2 model does not match the observed data at all and is excluded by the polarimetry method. However, the presence of mixed term could not be explained by either magnetic model within the approximation of a single-domain sample. This term may originate from the overlap of scattering intensities from different domains present in the crystal. Two possible types of domains can exist in our single crystal. The first possibility is the presence of structural twin domains. In the case of the (h0l) plane, two structural twins are related by a four-fold rotation matrix about the c-axis. The incorporation of such structural twins in our calculation does not affect the mixed term at all. The other possibility is the presence of multiple magnetic domains such as configuration K-domains and orientation Sdomains [START_REF] Brown | Magnetic structure studied with zero-field polarimetry[END_REF]. The scattered intensity at any particular Q is not affected by the presence of multiple K-domains. Therefore, these were not included in our calculations. Orientation S-domains occur whenever a magnetic structure has lower symmetry than the configurational symmetry. In the present case, two S-domains are created by the x -z mirror plane. However, the inclusion of these S-domains also did not produce the mixed term.

Another possible origin of the mixed term could be the tilting of magnetic moments away from the b-axis into the ac-plane. Our test calculations indeed confirmed that the magnetic interference term can be simulated only if the magnetic moments are slightly tilted away from the b-axis into the ac-plane. In terms of symmetry, the tilting of magnetic moments toward the ac-plane is not restricted and feasible with Γ(3) representation (hereafter denoted as the G3 model), which can be represented by the linear combination of G1 and G2 modes. Magnetic moments for the Ni and Nd sites obtained from the basis function calculated for the G3 mode resolved along the crystallographic axes as (u, v, w) where u, v and w are free parameters to be determined with refinements. The G3 model fits the data significantly better as presented in Fig. 3(a) while the refined structure is plotted in Fig. 3(b). All the least-squares refinement results are presented Table III. With the G3 magnetic model, the calculated polarization matrices for the satellite reflections at Q1 and Q2 are in good agreement (χ 2 = 7%) with the observed polarization matrices. In the final step, the volume fractions of S-domains were refined to obtain the best fit (χ 2 = 4%) with the observed data. Fitted matrices are presented in Table II.

The proposed G3 magnetic model satisfactorily describes the observed magnetic intensities, however, the refined magnetic parameters may contain significant errors since the integrated intensities used in the refinements were obtained from typical ω-scans, and due to diffuse nature of the magnetic peaks these integrated intensities have systematic errors. Moreover, the magnetic form factor of Ni 2+ ions is only used in the refinements and assumed as isotropic, which is not the case as Ni 3+ ions are also expected to be present in the sample. Furthermore, due to the covalency effect, the magnetic form factor can be strongly anisotropic. In addition, the number of structural and magnetic reflections used in the least-square refinements and in the polarization analysis is low. Therefore, the absolute values of the magnetic moments obtained may contain significant errors. Nevertheless, our measurements strongly suggest the development of a unique magnetic order in the oxygen-doped Ln2NiO4+δ system in the highly-doped regime. The interstitial oxygen atoms play a crucial role in the stabilization of such magnetic order along the c-axis along which the magnetic unit cell is doubled.

Temperature dependence of the magnetic structure

The temperature evolution of observed matrix elements is plotted in Fig. 3(c) for the Q1=(0.64, 0, 1.5) reflection in the temperature range of 2-20 K. The P(x,x) element remains constant over the investigated temperature range while both P(y,y) and P(z,z) components increase rapidly with increasing temperature mostly after TN1=8 K. The P(y,z) term simultaneously decreases and goes to 0 around 20 K. Referring to our discussion in the previous section, the observed matrix elements at 20 K, as shown in Table II for (0.64, 0, 1.5), demonstrate that the spins are directed along b-axis with a very small component along the ac-plane. Therefore, a significant part of the accomponent of Ni-moments is lost in the rearrangement process with increasing temperature beyond TN1. Such a temperature evolution certainly indicates a reorientation of Ni 2+ /Ni 3+ spins between 2-20 K which possibly occurs at 16 K according to our magnetic susceptibility data (see Fig. 1(e)). We should emphasize that the Nd 3+ ions follow the same magnetic representation as to the Ni-ions. Thus, the Nd 3+ ordering at TN1 is not expected to induce any spin reorientation. Therefore, such a transition in the Ni sublattice is mainly triggered by short-range magnetic correlations in the Nd sub-lattice which possibly exists at T ≥ TN1.

Magnetic field-induced transition

The magnetic field dependence effect on the magnetic Bragg peaks was studied by neutron diffraction below 10 K with applied magnetic field Hap ∥ c-axis. In Fig. 3(d), we plot the influence of the external magnetic field on the intensity of magnetic satellite at (0.64, 0, 2.5) at different temperatures. Upon application of the magnetic field ∥ caxis i.e. perpendicular to the NiO2-planes, a strong decrease in the peak intensity is observed with Hap ≥ 3.5 T and T ≤ 8 K, in fine agreement with our fielddependent isothermal magnetization data. The peak intensity decreases by 40% upon the application of magnetic field of 5.5 T at 2 K. The l-scan profiles of the magnetic peak is presented in the inset of Fig. 3(d) for Hap = 0 and 5.5 T. We note that the maximum intensity change is observed for this satellite as a function of magnetic field. This particular magnetic peak is weak compared to the other satellites, and appears only below T ≈ TN1 and then rapidly increases with decreasing temperature down to 2 K. Such an evolution manifests its correlation to the ordering of Nd 3+ ions. Therefore, the field-induced effect is possibly due to the strong contribution of Nd 3+ magnetic moments on the magnetic structure factor of this magnetic satellite, and the rotation of Nd 3+ magnetic moments under the magnetic field. Thus, our neutron diffraction data together with magnetization measurements reveal a strong magnetic field induced effect of Nd2NiO4.23, however, our present neutron diffraction data is not enough for quantitative analysis of such changes in the magnetic structure.

IV. CONCLUSIONS

In summary, the crystal structure of Nd2NiO4.23 is revealed to be three-dimensionally modulated, owing to the long-range ordering of excess oxygen atoms in the investigated temperature range of 2-330 K. The oxygen ordering, essentially established within the ab-plane, also involves an important electronic coupling along [001], as it directly strengthens the bonding throughout the Nd2O2layer. The important consequence then is the establishment of a magnetic exchange path along the Ni-Oap-Oint-Oap-Ni atoms, the Ni-Ni distances between two layers being 6.75 ˚A and 7.28 ˚A. An enhanced magnetic correlation between the NiO2 planes is also indicated by a doubling of the magnetic unit cell along the c-direction. The antiferromagnetic order of the Ni-sublattice is evidenced below TN ≈ 150 K while the magnetic order of Nd 3+ is detected below 8 K. The incommensurate magnetic structure is characterized by the magnetic satellites at (h ± ε, 0, l/2) with h and l being odd integer numbers with magnetic incommensurability, ε ≈ 0.36. The observed magnetic ordering is thus substantially different compared to the stripe spin order usually observed in hole-doped RP-Nickelates following the RE substitution by Sr. Thereby the incommensurate component of the structural modulation vector is essentially in the ab-plane, while the incommensurate magnetic component points along the c-axis. Thus, the highly oxygen-doped Nd2NiO4.23 presents an interesting scenario in which long-range incommensurate ordering of interstitial oxygen atoms, and a long-range incommensurate antiferromagnetic order coexists down to 2 K. We reveal that the magnetic correlation along the c-axis is mediated through the excess oxygen atoms at the interstitial sites, establishing a stronger magnetic correlation between the NiO2 planes, and thereby doubling the magnetic unit cell along this direction. This is certainly a promising perspective as it would allow to achieve complex magnetic ordering, as a consequence of long-range oxygen ordering, the latter can then be tuned by the excess oxygen stoichiometry.

FIG. 1 .

 1 FIG. 1. (a) Idealized layered crystal structure of oxygen-doped Nd2NiO4.23, consisting of interspersed NiO6 octahedral and Nd2O2 rock salt type layers. The excess oxygen atoms (Oint) occupy the tetrahedral ( , , ) sites coordinated by four apical oxygen atoms. Please note that on average only one interstitial oxygen atom is present per unit cell. (b) A section of the unit cell between 0 ≤ z ≤ 0.5 displaying possible magnetic exchange paths between the Nd2O2 layers through the Nd and excess oxygen atoms. The arrows represent the displacement direction of apical oxygen atoms due to Coulombic repulsion by oxygen interstitials. (c) An excerpt of the Rietveld fit of the room temperature synchrotron X-ray powder diffraction data. The inset represents a profile fit of the oxygen superstructure reflections with modulation vectors Qn = ±0.813a * ±0.519b * .The details of indexation process can be found elsewhere[START_REF] Maity | Large-scale oxygen order phase transitions and fast ordering kinetics at moderate temperatures in Nd2NiO4+δ electrodes[END_REF]. (d) Refined anisotropic thermal ellipsoids of the apical (Oap) and equatorial oxygen (Oeq) atoms demonstrating strong structural disorder along the ab-plane and the caxis, respectively. The refined bond lengths shown here are obtained from an average structure refinement while the real structure is strongly distorted, corresponding to a tilt of ca. 25 • for the NiO6 octahedra and a shift of about 2 ˚A of apical oxygen atoms towards[110] with Oint-Oap distances of around 2.8 ˚A instead of 2.13 ˚A for the non-distorted tetrahedron[START_REF] Ceretti | Low temperature oxygen diffusion mechanisms in Nd2NiO4+δ and Pr2NiO4+δ via large anharmonic displacements, explored by single crystal neutron diffraction[END_REF][START_REF] Maity | Large-scale oxygen order phase transitions and fast ordering kinetics at moderate temperatures in Nd2NiO4+δ electrodes[END_REF]. (e) Temperature dependent DC magnetic susceptibility curves of the Nd2NiO4.23 single crystal recorded with an applied magnetic field of 1 T parallel to the c-axis and the ab-plane. The inset represents a first-order temperature derivative of the inverse magnetic susceptibility data. (f) Isothermal magnetization curves measured at 2 K and 300 K with magnetic fields applied parallel to the c-axis and the ab-plane. (g) First-order field derivative of the magnetization data shown in (f).

FIG. 2 .

 2 FIG. 2. (a) Neutron diffraction (h0l) reciprocal space plane map of the Nd2NiO4.23 single crystal obtained at 2 K on DMC at SINQ, PSI (λ = 2.458 ˚A). Al denotes powder diffraction rings from the Aluminum sample holder. (b) and (c) One-dimensional line cuts obtained from the T-dependent plane maps. The scans are for the (0.64, 0, 1.5) magnetic satellite along the reciprocal h and l directions. A constant offset value is applied along the y-axis for clarity. (d)The temperature evolution of integrated intensity of the (0.64, 0, 1.5) magnetic satellite. (e) ω-scan peak profiles obtained for (0.64, 0, 1.5) reflection in the spin-flip (x,-x) and non-spin-flip (x,x) channels at 2 K on TASP at SINQ, PSI using MUPAD.

FIG. 3 .

 3 FIG. 3. (a) Plot of calculated and observed magnetic structure factors obtained using the G3 magnetic structure shown in (b). (b) The proposed magnetic structure of Nd2NiO4.23 at 2 K. (c) Temperature evolution of polarization matrix elements for the (0.64, 0, 1.5) reflection. Matrices were recorded with MUPAD on TASP at SINQ, PSI. (d) Temperature evolution of the (0.64, 0, 2.5) magnetic peak intensity with different applied magnetic fields which were applied parallel to the crystallographic c-axis. Measurements were performed at RITA II at SINQ, PSI (Ei = Ef = 3.4 meV). The inset shows the magnetic field dependence of the corresponding reflection at 2 K. Solid lines are only guide to the eyes.

TABLE I .

 I Magnetic parameters obtained from the fitting of ZFC magnetic susceptibility data with the Curie-Weiss law in the temperature range of 150-330 K.

	h r.
	Crystallographic axis	χ0(10 -3 × emu.mol -1 )	(emu.mol.K)	Θ (K)	µeff (µB)
	[001]	0.86	3.89	-54.2	5.57
	[110]	2.42	2.74	-31.2	5.87

[ l. u. ]

ACKNOWLEDGMENTS

This work is based on experiments performed at Paul Scherrer Institute (PSI), Villigen, Switzerland. The authors acknowledge the beam times used on DMC, TASP, ZEBRA, EIGER and RITA-II instruments at PSI/SINQ as well as laboratory equipment and support from LDM/PSI and the "Plateforme d'Analyse et de Caract´erisation" of the ICG. Montpellier. The authors gratefully acknowledge the financial support from grant 200021L 157131 and the French National Research the Swiss National Science Foundation (SNF) through Agency (ANR) through grant 14-CE36-0006-01 of the SECTOR project