Local lattice distortions and electronic phases in perovskite manganite Pr0.5Sr0.5MnO3

We use variable temperature and magnetic field total x-ray scattering to study the crystal structure of the strongly correlated Pr0.5Sr0.5MnO3 perovskite, which is a paramagnetic insulator at room temperature, becomes a ferromagnetic metal at 272 K and, upon further decreasing the temperature, turns into an antiferromagnetic insulator at 105 K. We find that a model featuring a monoclinic symmetry captures the structure and its temperature and field evolution well, eliminating the need to evoke a phase segregation scenario as done in prior studies. It appears that coupled variations in Mn–oxygen bonding distances and angles from their values in an undistorted perovskite lattice, i.e., coupled local lattice distortions, assist the phase transitions in Pr0.5Sr0.5MnO3, contributing to its unique physical properties. Local structural distortions thus emerge as an important degree of freedom in strongly correlated systems, in particular perovskite manganates, and, therefore, they should be fully accounted for when their fascinating physics is considered.


Introduction
Due to the interaction between electronic, magnetic, and lattice degrees of freedom, strongly correlated systems are known to host complex electronic phases.Among these systems, particularly prominent are RE 1−x A x MnO 3 perovskites (RE is rare earth ion and A is alkaline earth ion) where non-trivial superconductivity, charge and orbital ordering (OO), metal-insulator transitions, and colossal magnetoresistance are observed [1][2][3][4][5].An archetypal representative of these perovskites is Pr 0.5 Sr 0.5 MnO 3 where Pr 3+ and Sr 2+ ions occupy the cavities between corner-sharing Mn-O 6 octahedra, as shown in figure 1(a).Above room temperature, the material is a paramagnetic insulator (PMI).Upon decreasing temperature to 272 K, it undergoes a second-order phase transition and becomes a ferromagnetic metal (FMM).Upon further decreasing the temperature to 105 K, a first-order transition takes place, and the material becomes an antiferromagnetic insulator (AFMI).Concurrently, charge and orbital orders (CO/OO) emerge [6][7][8][9][10].The sequence of phase transitions is illustrated in figure 1(b) [10][11][12][13].Notably, upon the application of magnetic field at a fixed temperature just below 105 K, the orders can be virtually 'melted' and the FMM phase recovered via a metamagnetic phase transition (figure 1(c)).
The rich phase diagram of Pr 0.5 Sr 0.5 MnO 3 is thought to arise from the presence of Mn species with different both valance states, such as, for example, Mn 3+ and Mn 4+ species, and perfection of the oxygen octahedra that enclose them.In particular, the 3d electrons of Mn 3+ are considered to adopt a t 2g 3 e g 1 electronic configuration, whose degeneracy is lifted by a distortion of the Mn 3+ -O 6 octahedra, leading to the appearance of short and long Mn-O bonds in the ac plane of the crystal lattice [14,15].On the other hand, the 3d electrons of Mn 4+ ions are considered to adopt a t 2g 3 e g 0 electronic configuration and, hence, form much less distorted Mn 4+ -O 6 octahedra where all Mn-O bonds have a similar length, as shown in figure S1 [supplementary material].Essentially, the (dis)appearance of short and long Mn-O bonds signals changes in (a) Perovskite network of Mn-O6 octahedra in Pr0.5Sr0.5MnO3where Pr and Sr species occupy the cavities between the octahedra.Mn atoms are in light brown, oxygen atoms are in blue and Pr/Sr atoms are in dark brown.The octahedra are both rotated and tilted with respect to each other such that the rotation Mn-Oeq-Mn (Φ) and tilt Mn-Oap-Mn (α) angles both appear smaller than 180 • .Given the chosen orientation, the short and long Mn-Oeeq bonds referred to in the paper appear in the (ac) plane while the Mn-Oap bonds appear in the (ab) plane.(b) Temperature evolution of the magnetization for given magnetic fields each shown by the respective data set.Branches of the curves showing data obtained on cooling and warming are given in blue and red, respectively.The data show that, upon cooling, the material undergoes a second-order PMI to FMM phase transition at 272 K and first-order FMM to AFMI transition at 105 K. (c) Magnetic hysteresis curves measured at 300 K and 95 K, where the material is in its PMI and AFMI state, respectively.Upon increasing the field at 95 K, the material undergoes a metamagnetic AFMI to FMM phase transition.(d) X-ray and (e) atomic PDF intensity color maps, where the intensity increases as the color increases from violet to dark red.Arrows point to changes in the x-ray patterns and PDFs occurring at the temperatures where the PMI to FMM and FMM to AFMI transitions take place.
the occupancy of e g orbitals, i.e. valance state of Mn species, thus reflecting the coupling between the electronic and lattice degrees of freedom in Pr 0.5 Sr 0.5 MnO 3 .To describe the coupling between the electronic and magnetic degrees of freedom, a few competing mechanisms have been evoked.In the case of the FMM phase, the coupling has been described in terms of the double exchange (DE) mechanism [16,17] considering that the charge transport is due to an exchange (hoping) of itinerant Mn e g electrons between nearby Mn ions through a e g (Mn)−2p σ (O)−e g (Mn)-type orbital hybridization while the ferromagnetism arises from a strong Hund's coupling between localized t 3 2g and itinerant e g electrons on the same Mn site, leading to a ferromagnetic alignment of the spins of nearby Mn atoms [18,19].In the case of the AFMI phase, the AFM order emerging at low temperatures has been described in terms of the super-exchange (SE) mechanism, featuring an interaction between localized t 2g electrons of nearby Mn atoms through a t 2g (Mn)−2p π (O)−t 2g (Mn)-type orbital hybridization.Lastly, the charge ordering (CO) accompanying the AFM order is believed to feature an alternate arrangement of Mn 3+ and Mn 4+ ions on the vertices of the perovskite lattice while the concurrent OO is associated with the ordering of e g orbitals of the former and emergence of an arrangement of alternating long and short in-plane (equatorial) Mn-O bonding distances [20][21][22][23].Notably, both the DE and SE mechanisms envisage a more or less complete Mn 3+ /Mn 4+ charge disproportionation, including the presence of distinct Mn 3+ -O 6 and Mn 4+ -O 6 octahedra in Pr 0.5 Sr 0.5 MnO 3 .However, a number of experimental and theoretical studies [24][25][26][27][28][29][30] indicated that in an addition to the prevailing O 2− valence state, a portion of oxygen atoms in RE 1−x A x MnO 3 perovskites are likely to be in a monovalent O 1− state, where the latter accommodate a 2p electron vacancy, referred to as an O 2p hole, in their outer shell.The studies also showed that, largely, O 2p holes and not Mn 3d electrons are current charge carriers in these materials.An itinerant electron model (IEO) picture was suggested, where, in the FMM phase, an O 2p electron with a constant spin hops from an O −2 anion to the O 2p hole of an adjacent O 1− anion with a Mn cation acting as an intermediary.Then, because of the strong hybridization between Mn 3d and O 2p orbitals, the spins of nearby Mn atoms align in parallel to each other, leading to the emergence of FM order.Thus, the IEO mechanism envisages that the presence of O 2p holes and their strong hybridization with Mn 3d orbitals and not Mn 3d electrons determine the conductivity and magnetic exchange pathways between Mn atoms in Pr 0.5 Sr 0.5 MnO 3 .Notably, the IEO mechanisms envisages the presence of little, if any, Mn 3+ /Mn 4+ charge disproportionation in Pr 0.5 Sr 0.5 MnO 3 .
An issue of a considerable importance, both for understanding the underlying physics and for potential applications, is determining the structural changes accompanying the observed sequence of phase transitions.Here we show that marked variations in local lattice distortions taking place during the phase transitions in Pr 0.5 Sr 0.5 MnO 3 , be they induced by changing temperature or magnetic field.Lattice distortions are indeed expected to be significant in this material because of the significant difference in the size of Pr +3 (1.179 Å), and Sr +2 ions (1.31 Å), leading to the emergence of local strain fields.Distortions in Mn-O 6 octahedra are also inherent to Pr 0.5 Sr 0.5 MnO 3 due to charge localization in the PMI and AFMI phases.The contribution of local lattice distortions to the rich physics exhibited by Pr 0.5 Sr 0.5 MnO 3 , however, has been paid little attention so far because prior studies have largely concentrated on the average crystal structure alone.In particular, the high-and low-temperature crystal structure of Pr 0.5 Sr 0.5 MnO 3 has been described in terms of high symmetry tetragonal space group (S.G.) I4/mcm and orthorhombic S.G.Fmmm lattices, respectively, and furthermore, a macroscopic phase segregation over an extended temperature range has been assumed to occur [10,[31][32][33][34][35][36].To fill the knowledge gap, we conducted total x-ray scattering experiments over a broad temperature range, including the PMI to FMM and FMM to AFMI phase transitions.We also conducted experiments under variable magnetic field.We employed a monoclinic S.G.P2 1 /m model [14,37,38] to account for the lattice distortions [39][40][41].This approach allowed us to describe the distinct phases of Pr 0.5 Sr 0.5 MnO 3 in terms of a unique structure model, without the need to evoke phase segregation.

Sample preparation
Polycrystalline Pr 0.5 Sr 0.5 MnO 3 sample was prepared using a conventional two-step solid-state route.For the synthesis, stoichiometric amounts of Pr 2 O 3 , MnO 2, and SrCO 3 fine powders were thoroughly mixed and calcined twice at 1000 • C with intermediate grinding.To achieve phase purity, the calcined powders were sintered at 1400 • C. The phase purity of the calcined sample was confirmed using in-house powder x-ray diffraction (XRD).

Magnetic properties characterization
The magnetic properties of Pr 0.5 Sr 0.5 MnO 3 were characterized on a physical property measuring system (PPMS) from Quantum Design.Experimental data for the magnetization taken at a constant field as a function of temperature are shown in figure 1(b).Upon decreasing temperature, the magnetization shows a steep upturn at T c = 272 K, reflecting the PMI to FMM transition.Upon further decreasing the temperature, the magnetization drops at T N = 105 K, signaling the emergence of AFMI order.Pronounced hysteresis, i.e. divergence between the data set obtained upon cooling and warming is seen in the temperature range between T c and T N , reflecting the presence of FM order between these temperatures.The hysteresis diminishes, remaining visible only in the vicinity of T N with the increase in the magnetic field, and finally disappears when that field approaches 3 T.As discussed in [12], the effect reflects the first order nature of the FMM-AMFI transition.Note that, as observed in other studies [7,10,19], the magnetization does not become zero in the PMI and AFMI phases due to the presence of FM clusters giving rise to a Griffiths-like behavior.Experimental data for the magnetization taken at 300 K and 95 K, i.e. at constant temperature, as a function of magnetic field are shown in figure 1(c).The former is typical for PM materials harboring FM clusters.The latter is characteristic to metamagnetic phase transitions where, upon the application of a strong magnetic field, an FM order is induced, and the magnetization reaches saturation.The critical field for inducing FM order in Pr 0.5 Sr 0.5 MnO 3 at 95 K appears close to 3.5 T.

Total x-ray scattering experiments
Synchrotron XRD experiments were performed at the beamline 28-ID-1 at the National Synchrotron Source-II, Brookhaven National Laboratory, using x-rays with energy of 74.46 keV (λ = 0.1665 Å).The powder sample was sealed in a Kapton tube and positioned inside a liquid He cryostat used to control temperature.The scattered x-ray intensities were collected with a PerkinElmer area detector in the temperature range from 300 K to 10 K in steps of 5 K. Two data sets were obtained at each temperature point.One of the data sets was obtained with the detector positioned 1000 mm away from the sample to achieve high resolution in reciprocal space necessary for Rietveld analysis of the average crystal structure.The other data set was collected with the detector positioned 204 mm away from the sample to reach wave vectors, q, (q max = 28 Å −1 in the current experiment) necessary to obtain high-resolution atomic pair distribution functions (PDFs).The latter were derived from the diffraction patterns using standard procedures [42].Intensity color maps of the experimental high-q max XRD patterns and PDFs derived from them are shown in figure S2 [supplementary material].Selected areas of the maps, indicating the presence of structural phase transitions with changing temperature, are shown in figures 1(d) and (e), respectively.Changes in the structure and, hence, physical properties of Pr 0.5 Sr 0.5 MnO 3 were also probed by using magnetic field as an external perturbation.For the purpose, experimental XRD patterns were collected at a fixed temperature of 95 K while applying a magnetic field with a strength of up to 5 T. Representative patterns are shown in figure S3 [supplementary material].

Average crystal structure as a function of temperature
To reveal the temperature evolution of the average crystal structure, we subjected the high-q resolution XRD data to Rietveld analysis using the software FullProf [43].Initially, we tested the suggested in literature tetragonal S.G.I4/mcm [1] S1; [supplementary material].A mixture of the two models would improve the overall fit factor [1,7] but not reveal accurately the complexity of the material's local structure.Therefore, we decided to employ previously suggested monoclinic S.G.P2 1 /m models for RE 1−x A x MnO 3 perovskites [14,38], which, contrary to the tetragonal and orthorhombic models, features distinct A (A = Pr, Sr) and Mn atom sites, providing an opportunity to account for the likely local structural distortions in Pr 0.5 Sr 0.5 MnO 3 arising from the different size of Sr 2+ and Pr +3 ions [14,37,44,45].One of the suggested monoclinic models has been derived from an orthorhombic S.G.Pnma structure where the rotation and tilting of Mn-O 6 octahedra with respect to each other may be described in terms of the (a − b + a − ) Glazer scheme [14,46].The model has been used to fit diffraction data for La 0.5 Ca 0.5 MnO 3 with success [38].The other suggested monoclinic model has been derived from an orthorhombic S.G.Imma structure where the rotation and tilting of Mn-O 6 octahedra with respect to each other may be described in terms of the (a − b o a − ) Glazer scheme [14,46].The model has been used to fit diffraction data for Pr 0.5 Sr 0.5 MnO 3 with success [14,38].To keep the number of refinable parameters as small as possible, structural constraints of the type suggested in [14,38] were imposed on both models.A comparison between the refinable structure parameters for S.G.I4/mcm, S.G.Fmmm and both monoclinic models is given in table S1 [Supplementary material].The monoclinic (a − b o a − ) model was found to consistently outperform the (a − b + a − ) one, as discussed in SI [figure S8 and related text].Therefore, we concentrated on the former and hereafter show structural parameters resulted from its refinement.Representative Rietveld fits based on the (a − b o a − ) S.G.P2 1 /m model are shown in figures 2(d)-(f).The fits are superior to those based on the tetragonal and orthorhombic models (compare with data in figures 2(a)-(c) and S5 [supplementary material]).Refined monoclinic angle, lattice parameters and unit cell volume from Rietveld fits are summarized in figure S6 [supplementary material].However, figure 3 (a-c) reflects the refined lattice parameters, monoclinic angle and unit cell volume computed from PDF fits.

Average crystal structure as a function of magnetic field
Representative Rietveld fits to XRD data obtained while varying magnetic field are shown in figure S3 [supplementary material].The fits feature the (a − b o a − ) S.G.P2 1 /m model found useful in analyzing the XRD data obtained while varying temperature.Rietveld refined values for the rotation angle Φ (see figure 1(a)), lattice parameters and unit cell volume are summarized in figure 4. Also, shown in the figure is the field evolution of a superlattice XRD peak [14,37,38] arising from the CO/OO order in the AFMI phase.The peak is seen to gradually disappear with increasing magnetic field, demonstrating the destruction/melting of the order during the metamagnetic transition from the AFMI phase to the FMM phase.

Local crystal structure as a function of temperature
To reveal the temperature evolution of the local crystal structure, we fit the experimental PDFs with the monoclinic (a − b o a − ) S.G.P2 1 /m model using the software PDFGUI [42].Representative PDF fits are shown in figures 5 and S7 [supplementary material].As it may be expected, it outperformed the S.G.I4/mcm and

Discussion
To understand the local atomic structure in Pr 0.5 Sr 0.5 MnO 3 perovskite, we use those in orthorhombic PrMnO 3 and cubic SrMnO 3 as reference points.In the latter, all Mn species are believed to be in the Mn 4+ valence state and Mn 4+ -O 6 octahedra appear undistorted, rendering all Mn-O bonds close to 1.88 Å.On the other hand, in PrMnO 3 , all Mn species are believed to be in the Mn 3+ valence state and Mn 3+ -O 6 octahedra appear Jahn-Teller (JT)-like distorted.Accordingly, the octahedra exhibit short (1.91 Å) and long (2.21 Å) equatorial (eq) Mn 3+ -O eq (ac plane; see figures 1 and S1) bonds.The apical Mn 3+ -O ap (ab plane; see figures 1 and S1) bonds appear close to 1.95 Å [47,48].For reference, typically, Mn 2+ -oxygen bonds are about 2.05 Å-2.15 Å in length [49][50][51].As our structure data show (figure 3(d)), at room temperature, the longest and shortest Mn-O bonds in Pr 0.5 Sr 0.5 MnO 3 appear close to 2.04 Å and 1.95 Å, respectively (figure 3(d)).Within the framework of DE/SE mechanisms, the PMI phase of Pr 0.5 Sr 0.5 MnO 3 would comprise distinct Mn 4+ and Mn 3+ sites, where the latter and the former would be undistorted and JT-like distorted, respectively.The absence of Mn-O bonds as long as 2.21 Å in Pr 0.5 Sr 0.5 MnO 3 shows the absence of fully distorted Mn 3+ -O 6 octahedra in the material.The absence of Mn-O bonds as short as 1.88 Å shows that the material does not comprise completely undistorted Mn 4+ -O 6 octahedra either.The observations indicate that, in its PMI phase, Pr 0.5 Sr 0.5 MnO 3 does not exhibit a complete Mn 4+ /Mn 3+ charge disproportionation.That is because, likely, the increased hole doping arising from the substitution of Sr 2+ for Pr 3+ , leads to an increased creation of O 2p holes through electron transfer to nearby Mn species, which largely equilibrates their valance states in Pr 0.5 Sr 0.5 MnO 3 [52].It also indicates that the material is not a phase mixture of PrMnO 3 and SrMnO 3 but a true solid solution where, in terms of degree of distortion, Mn-O 6 octahedra are different from those in the end members, a picture common to RE 1−x A x MnO 3 perovskites [44].While the PMI and AFMI phases show long, intermediate, and short Mn-O bonding distances consistent with charge localization, the FMM phase exhibits one type of Mn-O bond distances consistent with charge delocalization.Refined values for the rotation Φ and tilt α angles are shown in (e) and (f), respectively.When the monoclinic angle is fixed to 90 • , the temperature evolution of the data shown in (a), (c), (d), (e), (f) does not change significantly, but the PMI phase (light blue shaded rectangle) to FMM phase (light green shaded rectangle) transitions at 272 K and the FMM to AFMI phase (light red shaded rectangle) transition at 105 K (vertical broken lines) sharpen a bit, as seen in figure S4.
The electronic bandwidth, W, is often used in considering the electronic transport and magnetic properties of RE 1−x A x MnO 3 perovskites.Within the tight binding approximation, W is defined as where <Mn-O-Mn> is the average Mn-O-Mn bond angle and d MnO is the average Mn-O bond length [53][54][55][56].In the PMI phase, due to thermal excitations, the magnetic moments of Mn atoms in Pr 0.5 Sr 0.5 MnO 3 are randomly oriented and the charge carriers are heavily scattered by phonons [57].As our structure data show, upon decreasing temperature down to 272 K, d MnO diminishes fast while the changes in Φ and α are relatively small (figures 3(d)-(f)).This would increase W, facilitating the emergence of a metallic-like state, where charge is transferred through hopping of O 2p holes from O 2− to O −1 , which can also be viewed as hopping of delocalized electrons in an opposite direction.In turn, due to the strong O 2p-Mn 3d orbital hybridization, the charge transfer would facilitate the emergence of FM order.In the emerged FMM phase, Mn sites appear largely equivalent, as demonstrated by the fact that Mn-O bonds appear near uniform and approximately 1.94 Å in length (figure 3(d)).While the Mn-O bond length keep shortening and the equatorial (Φ) Mn-O-Mn angle keeps increasing towards 180 deg with decreasing temperature below 272 K, which favors the FMM state, the tilt (α) angle deviates further from 180 • , reducing the overlap between Mn 3d and O 2p orbitals and hence their hybridization, which destabilizes the FM exchange interaction between the spins of nearby Mn atoms.At 105 K, the exchange interactions lose their FM character and Pr 0.5 Sr 0.5 MnO 3 becomes an AFM insulator through a first order phase transition.Concurrently, Mn d x2−y2 type orbitals order along the a axis of the crystal lattice, which is seen to elongate below 105 K.In the process, the c axis shrinks.Overall, the unit cell volume increases (figure 3(c)), leading to a further diminished overlap between Mn 3d and O 2p orbitals and stabilization of the AFMI phase.The rotation angle Φ remains near constant and the tilt angle α increases with decreasing temperature below 105 K, which is also consistent with the stabilization of the AFMI phase.Similarly to the case of PMI phase where charge is localized, AFMI Pr 0.5 Sr 0.5 MnO 3 exhibits little but not negligible Mn 4+ /Mn 3 charge disproportionation, as demonstrated by the relatively small difference between the long (1.95 Å) and short (1.91 Å) Mn-O bonds (figure 3(d)).Indeed, the disproportionation and CO phenomena in Pr 0.5 Sr 0.5 MnO 3 have been suggested to feature Mn +3.5+δ and Mn +3.5−δ species, where, similarly to Mn 3+ ions, the latter occupy JT-like distorted oxygen octahedra while, similarly to hypothetical Mn 4+ ions, the former occupy near perfect oxygen octahedra [58].
As shown in prior studies [59][60][61], the delicate balance between competing FM and AF orders in the AFMI phase can be tipped in favor of the former upon the application of strong magnetic field.In line with these studies, our structure data show that the characteristic for the AFMI phase OO/CO order is destroyed by the application of magnetic field in the order of 3.5 T, as indicated by the disappearance of superlattice peak in the XRD data (figure 4(b)).Upon destroying the order, Mn d x2−y2 type orbitals misalign, the a lattice parameter shortens (figure 4(a)) and the unit cell volume diminishes (figure 4(d)).The later would lead to an increase in the overlap between Mn 3d and O 2p orbitals and, hence, increase in the electronic bandwidth W, thereby favoring the charge delocalization and FM order.Concurrently, the octahedral rotation angle Φ gets closer to 180 • (figure 4(c)), consistent with the emergence of a FMM phase.Representative data sets for the crystal structure in the PMI, FMM and AFMI phases of Pr 0.5 Sr 0.5 MnO 3 obtained through fits to experimental PDFs are summarized in table S2 [supplementary material].The importance of the data sets is that, contrary to the high symmetry models used so far, they describe the material in terms of a realistic monoclinic structure.As such, the data sets can be used as a structural basis for computations exploring likely mechanisms for charge transport and exchange interactions in Pr 0.5 Sr 0.5 MnO 3 and related doped manganites.

Conclusions
Our experimental study confirms that Pr 0.5 Sr 0.5 MnO 3 exhibits three distinct phases as a function of temperature.Distorted Mn-O 6 octahedra accommodating distinct Mn-O bonding distances are present in the material wherever it is an insulator, and the local octahedral distortion disappears in the metallic phase.The distortion is, however, smaller than the one envisioned in models invoking a distinct Mn 3+ /Mn 4+ charge disproportionation.
At atomic level, the phases can be described in terms of a perovskite lattice with a local monoclinic symmetry.The crystallographic symmetry is reduced locally because the lattice comprises different in size atomic species, i.e.Pr and Sr, filling the cavities between the octahedra, inevitably leading to the emergence of local lattice strain.The strain is accommodated by unique rotations and tilting of Mn-O 6 octahedra, including changes in Mn-O bonding distances and angles with changing temperature and magnetic field, that affect both the hopping integral of the charge carriers and character of exchange interactions between nearby Mn spins.Altogether, the phase transitions exhibited by Pr 0.5 Sr 0.5 MnO 3 appear to be greatly facilitated by the marked ability of the underlying perovskite lattice to accommodate a variety of local distortion patterns that favor particular electronic phases.Our results show that local lattice distortions are an important degree of freedom in strongly correlated materials such as perovskite manganates, and, therefore, should be fully accounted for when their fascinating physics is considered.

Figure 1 .
Figure 1.(a) Perovskite network of Mn-O6 octahedra in Pr0.5Sr0.5MnO3where Pr and Sr species occupy the cavities between the octahedra.Mn atoms are in light brown, oxygen atoms are in blue and Pr/Sr atoms are in dark brown.The octahedra are both rotated and tilted with respect to each other such that the rotation Mn-Oeq-Mn (Φ) and tilt Mn-Oap-Mn (α) angles both appear smaller than 180 • .Given the chosen orientation, the short and long Mn-Oeeq bonds referred to in the paper appear in the (ac) plane while the Mn-Oap bonds appear in the (ab) plane.(b) Temperature evolution of the magnetization for given magnetic fields each shown by the respective data set.Branches of the curves showing data obtained on cooling and warming are given in blue and red, respectively.The data show that, upon cooling, the material undergoes a second-order PMI to FMM phase transition at 272 K and first-order FMM to AFMI transition at 105 K. (c) Magnetic hysteresis curves measured at 300 K and 95 K, where the material is in its PMI and AFMI state, respectively.Upon increasing the field at 95 K, the material undergoes a metamagnetic AFMI to FMM phase transition.(d) X-ray and (e) atomic PDF intensity color maps, where the intensity increases as the color increases from violet to dark red.Arrows point to changes in the x-ray patterns and PDFs occurring at the temperatures where the PMI to FMM and FMM to AFMI transitions take place.
model with lattice parameters a = b = √2a p and c = 2a p , where a p is the cell parameter for a hypothetical perovskite structure with an undistorted cubic lattice.The model did not perform well, particularly at low temperatures figures 2(a)-(c).Values for the refined lattice parameters are summarized in figure S4(a) [supplementary material].The XRD data for the AFMI phase were also attempted with a previously suggested orthorhombic S.G.Fmmm (a = b = c = 2a p ) model, whose performance did not turn out better (see figure S5; [supplementary material].Values for the refined lattice parameters are summarized in figure S4(b) [supplementary material].The temperature evolution of the lattice parameters reflects the presence of PMI to FMM and FMM to AFMI phase transitions.Atomic positional parameters resulting from the models, however, have a limited value because the models neglect the presence of different A-type species and related to them local lattice distortions in Pr 0.5 Sr 0.5 MnO 3 (see table

Figure 2 .
Figure 2. fits (orange line) to XRD patterns (black circles) for Pr0.5Sr0.5MnO3based on tetragonal S.G.I4/mcm and (a − b o a − ) monoclinic P21/m type model structures.Vertical blue lines show the position of Bragg peaks.For clarity, the residual difference (green line) is shifted downward by subtracting a constant number.The values of the goodness of fit indicators Rwp are shown for all fits in (%).Also shown are the temperatures where the patterns have been collected.

Figure 3 .
Figure 3. Temperature evolution of PDF refined structural features for Pr0.5Sr0.5MnO3,using a (a − b o a − ) monoclinic S.G.P21/m model.The features include (a) lattice parameters, (b) monoclinic angle β, (c) atomic volume, and (d) Mn-O bonding distances.While the PMI and AFMI phases show long, intermediate, and short Mn-O bonding distances consistent with charge localization, the FMM phase exhibits one type of Mn-O bond distances consistent with charge delocalization.Refined values for the rotation Φ and tilt α angles are shown in (e) and (f), respectively.When the monoclinic angle is fixed to 90 • , the temperature evolution of the data shown in (a), (c), (d), (e), (f) does not change significantly, but the PMI phase (light blue shaded rectangle) to FMM phase (light green shaded rectangle) transitions at 272 K and the FMM to AFMI phase (light red shaded rectangle) transition at 105 K (vertical broken lines) sharpen a bit, as seen in figure S4.

Figure 4 .
Figure 4. (a) Temperature evolution of the (a) lattice parameters, (c) octahedral rotation angle Φ and (d) atomic volume for Pr0.5Sr0.5MnO3as obtained by Rietveld fits to XRD patterns collected at 95 K while increasing the magnetic field from 0 T to 5 T in steps of 0.5 T. The fits are based on a (a − b o a − ) monoclinic S.G.P21/m model Evolution of a CO/OO superlattice peak emerging in the XRD pattern of the AFMI phase is given in (b).The peak disappears with increasing field, signaling the 'melting' of CO/OO orders.Vertical broken line in (a), (c) and (d) marks the metamagnetic AFMI (light red shaded rectangle) to FMM (light green shaded rectangle) phase transition taking place at a critical magnetic field of about 3.5 T.

Figure 5 .
Figure 5. Fits (orange line) to experimental PDFs (black circles) for Pr0.5Sr0.5MnO3based on tetragonal S.G.I4/mcm and (a − b o a − ) monoclinic S.G.P21/m model structures.For clarity, the residual difference (green line) is shifted downward by subtracting a constant number.The values of the goodness of fit indicators Rwp are shown for all fits in (%).Also shown are the temperatures where at the PDFs have been obtained.For completeness, the same fits are shown over an extended range of r-values in figure S7.