Total X-ray scattering and big-box modeling of pressure-induced local disorder and partial amorphization in CsPbBr_3

Domain: earth_science
Published: 2025-08-16
Editorial outcome: major_revision

Abstract
The mechanisms governing pressure-induced amorphization and its reversibility in halide perovskites are not yet fully understood, particularly the contribution of local disorder. We performed high-pressure synchrotron total X-ray scattering and reverse Monte Carlo (RMC) big-box modeling using CsPbBr3 as a model system to investigate short-range structural evolution in both the ordered and partially amorphous phases. While diffraction data indicate that long-range order persists up to 2 GPa, pair distribution function (PDF) analysis reveals significant local distortions, including PbBr6 octahedral tilting and Cs displacement, which influence the band gap through a complex interplay between bond compression and angular tilting. Beyond 2 GPa, CsPbBr3 undergoes partial amorphization, with significant disordering of Cs and Br, while the Pb sublattice remains preserved, allowing for structural recovery upon decompression. Our work, accounting for both short- and long-range structural evolution through RMC modeling, successfully captures how disorder shapes the structural response of halide perovskites under pressure.

Introduction
Halide perovskites have emerged as a leading class of materials in optoelectronic applications, particularly in photovoltaic cells1. Since the development of the first perovskite-based solar cells in 2009 with a modest 3.8% power conversion efficiency, subsequent advancements have not only pushed efficiencies over 33% in tandem cells2 but also expanded their use into light-emitting diodes, photodetectors, lasers, and transistors3–5. Their remarkable success is due to their exceptional properties such as strong light absorption, long charge-carrier lifetimes, and low-cost synthesis. The ability to chemically tune their structure through replacing one or more chemical species allowed the synthesis of a wide variety of systems, fully inorganic or hybrid organic-inorganic, with tailored optoelectronic properties. Additionally, external factors like temperature and pressure can modulate their structures and properties. Multiple studies have investigated the behavior of halide perovskites under high pressure6–10, reporting a range of structural and electronic/optical phenomena, including crystalline phase transitions11,12, metallization13 and enhanced photoluminescence14. A common behavior observed in halide perovskites is the transition to a partially amorphous phase under pressure, with the onset of this transition varying across different systems10. Halide perovskites often exhibit the ability to recover their original structure upon decompression, highlighting the reversibility of their structural changes under pressure. While reversible pressure-induced amorphization is a known phenomenon, the fundamental mechanisms driving this structural recovery remain unclear. To address this, new methods beyond the conventional analysis of X-ray diffraction (XRD) patterns are required. A promising approach lies in focusing on the short-range structure and examining the material’s evolution on a more localized scale. While XRD primarily provides information on the average crystal structure, local structural behavior, i.e., on the Angstrom scale, can differ significantly. Indeed, this approach has proven particularly effective in uncovering local disorder in halide perovskites. For instance, in CsSnBr3, local off-centering of Sn atoms—undetectable by conventional crystallography—has been identified as the cause of asymmetries in the cubic phase15. Similarly, studying the system locally in (MHy)PbBr3 (MHy+=methylhydrazinium) has revealed how lone pair activity drives structural distortions, enhances electron-phonon interactions, and influences bandgap behavior and charge dynamics16.

The PDF, or G(r), is a powerful tool to study the local atomic structure of materials, as it captures the probability of finding a pair of atoms separated by a specific distance in real space17. By revealing local density variations and atomic arrangements, the PDF provides valuable insights into both local (short-range) and average (long-range) atomic structure18. Performing total X-ray scattering experiments makes it possible to extract the PDF and construct atomistic structural models which describe the local atomic environment at the unit-cell scale, known as the small-box approach. This method has proven effective in our previous work on Cs2AgBiBr6 at high pressure, where it revealed that a lower-symmetry model better describes the system on the local scale compared to symmetry observed through conventional XRD19. A more complex approach involves the use of a supercell built on the material’s unit cell which allows for more sophisticated models of the structure. Big-box modeling is usually coupled with the RMC method where a randomly selected atom is displaced by a random distance to improve the agreement between the modeled and experimental data20. It was initially developed to study liquid and amorphous phases21, however, this approach is also suitable for studying correlated disorder in crystalline materials, as it provides a detailed representation of the disordered structure. It incorporates local correlations consistent with the measured total scattering data while simultaneously reproducing the average structure observed in the Bragg profile22. This approach produces a ‘snapshot’ of the structure capturing both static and dynamic disordering processes, making it ideal to study pressure-induced modification and amorphization processes23.

Here, we employ total X-ray scattering combined with RMC big-box modeling to study the high-pressure behavior of CsPbBr3, a halide perovskite with a distorted perovskite structure typically assigned to the orthorhombic space group Pbnm (or Pnma) at room temperature24. The perovskite structure has the general formula ABX3, where A is a monovalent cation (either an inorganic ion or an organic molecule), B is a divalent metal cation, and X is a halide anion. CsPbBr3 combines a relatively simple inorganic composition with remarkable optical properties25, making it a model system for investigating pressure-induced structural and electronic changes, including reversible amorphization and the evolution of local disorder. Previous studies have noted a pressure-induced bandgap shift from red to blue around 1 GPa and subsequent partial amorphization at 2.4 GPa, with explanations ranging from isostructural phase transitions at 1 GPa to changes in Rashba splitting26–28. First-principles calculations indicate that the nonmonotonic pressure dependence of the bandgap results from competing effects of bond shrinkage and octahedral distortion, suggesting complex underlying mechanisms beyond straightforward structural changes29. Our study focuses on the evolution of local disorder under compression, employing a methodological approach that provides both detailed insights into the role of disorder within the crystalline phase and information about the amorphization process and its reversibility. We report structural modeling of the high-pressure partially amorphous phase of a halide perovskite based on experimental data, which has not been previously characterized in detail. While PDF studies have revealed local distortions in halide perovskites, most have relied on small-box modeling and focuses on temperature effects15,19,30–32, only two prior studies have applied big-box RMC simulations to investigate temperature-induced disorder23,33. We propose that the bandgap behavior under pressure is mainly driven by disorder effects and the reversibility of the amorphization stems from the preserved Pb sublattice, despite significant distortion of the PbBr6 octahedra and off-centering of the Cs atoms. Our study highlights the critical importance of understanding both the local and average structural configurations in halide perovskites to accurately capture the structure–property relationships.

Results
High-pressure powder XRD of CsPbBr3 (Fig. 1a) and the atomic PDFs (Fig. 1b), derived from total X-ray scattering data collected at a shorter sample-to-detector distance, were obtained in a diamond anvil cell (DAC) up to 5 GPa. We used silicone oil as a pressure-transmitting medium which remains quasi-hydrostatic up to 3 GPa, with pressure variations across the sample chamber less than 0.2 GPa at 5 GPa34. At ambient pressure, the diffraction pattern exhibits Bragg reflections characteristic of the Pbnm space group, as previously reported in powder diffraction studies35–38. While some recent high-resolution single-crystal studies have suggested monoclinic distortions or superstructures39,40, such features are typically not distinguishable in powder diffraction data, as also noted in the single-crystal study by Liu et al.40. Therefore, we adopted the orthorhombic Pbnm model for refinement and analysis, which yields stable and physically meaningful fits of our powder diffraction data (see Figures S1–S3). We performed Rietveld refinement on the XRD data up to 1.6 GPa to extract the lattice parameters (see Figure S6). Above 2 GPa, most peaks, particularly those above 4.5°, start to vanish or significantly broaden, signaling the onset of amorphization. Despite this, a few reflections remain sharp and detectable up to 5 GPa, indicating that partial long-range order is preserved in the structure up to this pressure value. This coexistence of amorphous and crystalline regions makes Rietveld refinement unsuitable beyond 2 GPa. Consequently, we switched to Le Bail fitting while maintaining the ambient pressure space group. We also attempted to fit the data using the recently proposed triclinic space group for CsPbBr3 at 2 GPa27, but at this pressure, the diffraction pattern is already too broadened for a meaningful distinction between low-symmetry space groups, and any could be fit using Le Bail refinement. The discrepancy with that study likely stems from differences in hydrostatic conditions and sample types (powder vs. single crystal). Focusing on the PDFs, at ambient pressure, our data show well-defined local correlations (e.g., Br–Cs, Br–Br, and Pb–Cs), consistent with an orthorhombically distorted perovskite lattice. This observation agrees with previous total scattering studies on related halide perovskites, which have shown that local distortions and octahedral tilting persist even when the average structure appears cubic or weakly distorted15,30,31. On increasing pressure, we observe short-range structural modifications even below 1.6 GPa, particularly in the 3.5 to 5.5 Å range (Fig. 1c), where primary contributions arise from the Br-Cs, Br-Br, and Pb-Cs first-neighbor distances. These changes reflect pressure-induced effects on the local structure that are not apparent in the average structure captured by XRD. Above 2 GPa, we detect significant alterations in both the shape and intensity of the peaks, consistent with the transition to a more disordered phase, as observed in the XRD patterns. The diminished periodicity of the structure can clearly be seen in the flattened G(r) at higher values of r.

Upon releasing the pressure, CsPbBr3 recovers its original crystalline structure, as highlighted by the comparison of the XRD data at ambient pressure before and after pressure release (Figure S7). The lattice parameters obtained through Rietveld refinement are almost identical to the ones of the sample before increasing pressure (Figure S8, Table S1). While this recovery demonstrates that the pressure-induced transition to the partially amorphous phase is mainly reversible, minor deviations from the initial pattern are still apparent. These include slight broadening and shifts in some Bragg peaks, suggesting residual strain within the crystal lattice. These subtle changes are also mirrored in the PDF comparison between the initial and post-release data (Figure S9), although the overall G(r) regains its original shape, differences in peak intensities and positions are evident, particularly in the 3.0-5.5 Å range. Such alterations highlight that, while the crystalline structure largely recovers, the local environment within the crystal lattice retains modifications induced by high-pressure conditions.

We analyzed X-ray PDFs using a Pbnm model in PDFgui, as shown in Figure S1018. However, since Pbnm is a relatively low-symmetry space group, this approach provided limited insight. Additionally, the refined lattice parameters became unstable and unphysical at higher pressures. Moreover, the single-unit cell model in PDFgui is likely too small to account for the increasing disorder beyond ~2 GPa. To address these challenges, we shifted to RMC simulations using RMCProfile21. We built a 6×6×4 supercell containing 2880 atoms, allowing for significantly greater variation in the atomic arrangements.

We started the RMC simulation by applying the distance windows constraint alongside the polyhedral constraint, which allows atoms to move as groups or units within a specified range. Given that CsPbBr3 adopts a perovskite structure at ambient conditions, we used the polyhedral restraint to fix the PbBr6 octahedra. Simultaneously, we fit XRD data using the parameters obtained from the Rietveld refinement, such as peak shape and scale factor. This method provided good fitting results for both the XRD and PDF data up to 1.2 GPa (Figures S11–S12). However, at 1.6 GPa, RMCProfile failed to fit the diffractogram, causing the PDF fit to fail as well. By excluding the average structure data from the fitting procedure, we successfully obtained a good fit for the local structure at this pressure. Beyond 2 GPa, the polyhedral constraint could no longer be applied, as the PbBr6 octahedra become too distorted to be meaningfully constrained to reference bond lengths and angles (Figures S13–S14). In response, we tested alternative approaches for data above this pressure, such as constraining only the Pb-Pb distances. This method yielded acceptable χ2 values for the total G(r) fit. However, further inspection of the partial PDFs showed that the Br-Cs and Br-Br distance distributions assumed unphysical shapes, with distances increasingly compressed to their lower limits at higher pressures. These unusual shapes suggest atomic arrangements with bond lengths that are too short to be physically meaningful (Figures S15–S16). The preliminary tests, briefly discussed here, are described in detail in the Supplementary Information. To overcome these challenges, we adopted bond valence sum constraints, which account for interatomic distances and atomic coordination41. This approach effectively fits the PDFs up to 5 GPa, providing physically consistent results (Figures S17–S24). It is worth noting that above 2 GPa, we used the parameters obtained from the Le Bail refinement to fit the Bragg data in RMCProfile. Since the Le Bail refinement does not provide a scale factor, which is required by RMCProfile, we estimated it by considering the scale factor obtained at 1.6 GPa and performing a least-squares fit between the observed and the theoretical peak intensities of the first main reflection, i.e., (110), for the diffractograms up to 5 GPa. To ensure accurate fitting, we first subtracted the background from experimental data using a Chebyshev function. Further on, all the results reported and discussed were obtained using bond valence sum constraints for Cs, Pb, and Br.

The total and the partial PDFs at 0.0 GPa, obtained through big-box modeling, are shown in Fig. 2a. The experimental PDF (black open circles) and the calculated PDF (red solid line) are in excellent agreement across the full r-range, demonstrating the suitability of the big-box approach for modeling the structural configuration at ambient conditions. The partial PDFs, representing contributions from specific atomic pairs (Pb-Pb, Pb-Br, Br-Br, Br-Cs, Pb-Cs, and Cs-Cs), display well-defined peaks corresponding to distinct interatomic distances, extending up to 23 Å. Focusing on the 3.3–5.6 Å range, the main contributions to the total PDF at these distances arise from the Br-Cs, Br-Br, and Pb-Cs first-neighbor interactions. Figure 2b–e illustrates the evolution of these partial PDFs with increasing pressure, specifically at 0.0 GPa (b), 0.4 GPa (c), 0.8 GPa (d), and 1.2 GPa (e).

With increasing pressure, the most significant changes are observed in the Br-Br peak (green curve in Fig. 2a–e), which broadens and shifts, indicating that pressure induces local distortions within the octahedra. Notably, the shape and intensity of the first peak in the G(r), primarily from the Pb-Br contribution, remain relatively unchanged up to 1.6 GPa (see Fig. 1b). This suggests that the PbBr6 framework is not significantly impacted within this pressure range. However, the Br-Cs and Pb-Cs partial contributions exhibit gradual broadening as pressure increases, reflecting enhanced disorder in the Cs local coordination environment. Pressure-induced changes in the Br–Cs and Br–Br partial PDFs resemble those observed in the orthorhombic γ-phase of CsPbI3, where pronounced changes were reported in the PDF between 3.5 and 6 Å, associated with I–I and Cs–I distances, while the Pb–I first-neighbor peak remained largely unaffected23. These similarities suggest that both thermal and pressure-induced disorder initially affect the A-site coordination and halide sublattice, while the metal-halide framework remains comparatively rigid within the same crystallographic phase.

At ambient pressure, CsPbBr3 exhibits nominal bond angles of 90° and around 175° for the intra-octahedral Br–Pb–Br bonds—slightly deviated from 180° due to octahedral tilting in the Pbnm phase—and ~160° for the inter-octahedral Pb–Br–Pb angles. These values are reflected in the angular distributions derived from the big-box modeled structure at 0.0 GPa, as shown in panels f and g of Fig. 2. It is worth noting that in (MA)PbI3 (MA=methylammonium) two distinct inter-octahedral angles are resolved at 10 K in the orthorhombic phase (~151° and ~162°), but they merge into a single broader peak centered near 160° above 100 K, consistent with the average angle we observe at room temperature33. Upon applying pressure up to 0.8 GPa, the intra-octahedral angle distributions broaden, indicating increased local distortion within the octahedra. Simultaneously, the distribution of inter-octahedral angles shifts toward 180°, suggesting a reduction in octahedral tilting. At 1.2 GPa, the inter-octahedral angle distribution shifts back toward 160° and broadens further, indicating the onset of more significant local disorder. This dynamic evolution of octahedral tilting correlates directly with the nonmonotonic bandgap behavior previously observed26,28 and is further confirmed by our absorption spectroscopy measurement, which we performed and analyzed under pressure (see Figure S25). Our measurements reveal a distinct trend: the bandgap narrows (red-shifts) up to ~1 GPa, followed by a reversal where it widens (blue-shifts) beyond 1.2 GPa. A previous experimental study attributed this behavior to an isostructural phase transition26. Shortening of the Pb–Br bond under pressure and reduced octahedral tilting (angles approaching 180°) enhance orbital overlap, leading to a bandgap reduction8,29. Beyond 1 GPa, the reintroduction of tilting (angles shifting back to ~160°) reduces orbital overlap, causing the bandgap to widen.

Our analysis of the XRD data and PDFs reveals no evidence of a structural phase transition between 0 and 1.6 GPa. Instead, we observe increasing local disorder, reflected in the broadening and shifting of both intra- and inter-octahedral angle distributions. The evolution of inter-octahedral angles, shifting toward 180° up to 0.8 GPa and returning to ~160° at 1.2 GPa, closely mirrors the observed bandgap changes. This suggests that the bandgap behavior may be driven by cumulative effect of local distortions within the PbBr6 framework, rather than a sharp phase transition.

At 1.6 GPa, both the partial PDFs, indicative of the bond lengths, and the angle distributions, broaden and shift, indicating a significant increase in disorder with pressure. As previously discussed, a much more pronounced change in the experimental data occurs at 2.0 GPa, indicating the onset of amorphization. This transition, clearly observed in the XRD data (see Fig. 1a), is also reflected in the big-box configurations, represented in Fig. 3 as snapshots of the 6×6×4 supercells for selected pressures along the c axis. In the ideal structure (the first snapshot in the sequence), the atoms are well-aligned, and the crystal symmetry is clearly evident. The configuration modeled with RMCProfile based on our experimental PDF at ambient pressure already exhibits some atomic displacements, primarily due to thermal effects. As pressure increases, the disorder becomes progressively more pronounced. Up to 1.6 GPa, it is still possible to discern the underlying crystal symmetry despite the noticeable distortions. However, at 2.0 GPa and beyond (3.4 and 5.0 GPa), the snapshots reveal severe deviations from the initial positions, particularly among the Cs atoms, which are significantly off-centered. These changes mark the transition to a highly disordered, amorphous-like state.

To gain deeper insights into the transition, we analyze the distributions of key bond lengths and inter- and intra-octahedral angles at pressures above 2.0 GPa. Figure 4 compares the partial PDFs for selected atomic pairs and the angular distributions for octahedral bond angles at pressures beyond the onset of amorphization. To better capture the structural evolution, we include data at 0, 0.8, and 1.6 GPa. The partial PDFs are shown for all three pressures to highlight the progressive changes leading up to amorphization, while the angular distributions are shown for 0 and 1.6 GPa to illustrate the evolution across the transition without overcrowding the plots.

The Pb-Pb PDF shifts to a shorter distance between 0 and 1.6 GPa, consistent with lattice shrinkage induced by pressure, and exhibits significant broadening and intensity reduction above 2.0 GPa. At higher pressures, the partial PDF develops a tail below the main peak, suggesting that Pb atoms occupy a broader range of positions due to lattice distortions and local strain. However, since RMCProfile does not enforce interatomic potentials, it is necessary to impose a limit on the shortest allowable atomic pair distances to prevent unphysical configurations21. For Pb-Pb, this limit was set at 5.1 Å at 0 GPa, based on the orthorhombic unit cell at ambient conditions, and reduced to 4.8 Å above 2 GPa to account for compressions effects, in line with the 4.6 Å Pb-Pb distance previously calculated for (MA)PbI3 at 6 GPa42. This constraint may influence the development of the tail. Interestingly, the position of the main Pb-Pb peak does not shift further after 2.0 GPa. In contrast, the Pb-Br PDF (Fig. 4b) remains sharp, despite a reduction in intensity when the amorphization begins at 2.0 GPa. On the other hand, the Br-Br distance distribution shows significant broadening and peak shifts (Fig. 4d), reflecting increasing distortions within and between the octahedra under compression. These observations suggest that while the PbBr6 octahedra are highly distorted, the Pb sublattice is relatively stable up to 5 GPa, as shown in the supercell snapshots in Figure S26 where only the Pb atoms are displayed.

Distortions of the octahedra are further reflected in the Br-Pb-Br (intra-octahedral) and Pb-Br-Pb (inter-octahedral) angular distributions (Fig. 4e and f, respectively). At ambient pressure, the intra-octahedral bond angles (e) exhibit peaks at 90° and around 175°, reflecting deviations from the ideal 180° due to octahedral tilting in the orthorhombic Pbnm phase, while the inter-octahedral bond angles (f) are centered around 160°, consistent with the expected tilt between adjacent octahedra. As pressure increases above 1.6 GPa, both angle distributions broaden, highlighting local deformations. Notably, the Br-Pb-Br distribution peaks shift to lower angles, while the Pb-Br-Pb develops a tail extending down to 100°, signaling severe tilting of the PbBr6 octahedra and significant lattice disorder. A pronounced low-angle tail down to 100° is not observed in any phase of (MA)PbI3 across the 10–400 K range, nor in (MA)PbBr3, including the orthorhombic phase at low temperature33,43. This appears to be a distinctive feature of a more disordered structural phase associated with amorphization under pressure.

To further quantify these distortions, we computed two statistical descriptors from the RMC configurations: the quadratic elongation 〈λ〉, which captures bond-length disorder, and the angular variance σ2θ, which reflects deviations from the ideal intra- and inter-octahedral angles (90° for Br–Pb–Br and 160° for Pb–Br–Pb, respectively). Both parameters remain nearly constant in the low-pressure regime and increases sharply between ~1.2 and 2.6 GPa, consistent with the onset of local distortion and the approach to the amorphization. While the Br–Pb–Br variance begins to rise slightly earlier, the Pb–Br–Pb variance reaches higher absolute values and shows a comparable trend at higher pressure. A similar angular variance metric was reported in a recent RMC study of (MA)PbI3, where σ2θ increases gradually with temperature from 10 to 400 K33. In contrast, our pressure-dependent σ2θ shows a much steeper rise beyond 2.0 GPa, highlighting the contrast between the gradual thermal broadening and the sharp pressure-induced angular distortion observed in CsPbBr3. As shown in Figures S27–S28, the pressure evolution of 〈λ〉 and σ2θ closely follows the bandgap trend, particularly the sharp increase above 1.6 GPa, supporting the interpretation that octahedral distortion plays a key role in modulating the optoelectronic response of CsPbBr3.

Regarding the Cs cations, we observe that increasing pressure induces significant broadening and intensity reduction in both the Pb–Cs (Fig. 4c) and Br–Cs (Figure S29e) partial PDFs, reflecting growing disorder in the Cs coordination environment, especially beyond 1.6 GPa. This suggests that Cs atoms are displaced from their ideal positions as pressure increases, contributing to the overall loss of crystallinity in the lattice. In CsPbI3, PDF and XRD studies have shown that Cs displacement increases with temperature within the orthorhombic Pbnm phase, even in the absence of a structural phase transition31. However, our data shows that compression drives this displacement much further, in parallel with pronounced octahedral distortions, ultimately leading to the collapse of long-range order. This highlights the cooperative role of both A-site and PbBr6 octahedral deformation in driving the pressure-induced amorphization of the perovskite lattice.

Another way to visualize the evolution of the system under compression, as obtained through RMC simulations, is to pack the supercell back into a single unit cell containing all atoms. Figure 5 shows the unit cells for pressures of 0.0, 0.8, 2.0, and 5.0 GPa. The configurations at remaining pressures are reported in Figure S30. These pressures shown in Fig. 5 were selected to represent key stages of the structural evolution on compression: the starting system at ambient pressure (0.0 GPa, Fig. 4a), the system under compression while still preserving long-range order (0.8 GPa, panel b), the onset of the amorphization (2.0 GPa, panel c), and the highest pressure measured (5.0 GPa, panel d).

At 0.0 GPa, the configuration presents atoms of each species overlapping which indicates a highly ordered crystalline lattice where the structural symmetry of CsPbBr3 is recognizable and the distribution of positions is due to thermal motion. At 0.8 GPa, we observe small but noticeable movements from the ideal sites, particularly among Cs and Br atoms, while the overall long-range order remains preserved. This intermediate state reflects the initial structural response to compression, with increased density as pressure is applied. By 2.0 GPa, significant atomic displacements become more evident, marking the transition to an amorphous-like phase. In particular, the Cs atoms are displaced from their ideal positions, and disorder in the vertices of the octahedra, i.e., the Br atoms, is also evident. These changes correspond to the broadening and shifting observed in the partial PDFs and angular distributions, highlighting the pronounced local disorder which starts to arise at this pressure. At 5.0 GPa, the system exhibits a highly disordered structure, with substantial overlap and dispersion of atomic positions, particularly among the Cs and Br atoms. However, the Pb atoms seem to still occupy the original sites, although they are statistically more dispersed. As previously mentioned, a few Bragg reflections remain detectable in the XRD patterns (see Fig. 1), indicating that the long-range order is not completely lost. These reflections can be indexed to the hkl planes associated with the Pb atoms (see Figure S37), suggesting that the Pb sublattice retains partial order even as the surrounding atoms are displaced.

Upon releasing the pressure, we observed the recovery of the initial phase, although with minor differences already discussed. This is a phenomenon well-documented in other halide perovskite materials, even when compressed to pressures exceeding 60 GPa13. Although we did not collect data at intermediate pressures, RMC simulations were conducted starting from the configuration obtained at 5 GPa. This approach enabled the successful recovery of more ordered configurations, beginning with the most disordered state observed at 5 GPa. Simulations conducted on the experimental data collected after the pressure release provide a large-box configuration that displays well-aligned atoms, with the structural symmetry clearly reestablished (Figures S38–40). Further analysis on the octahedral bond lengths and tiltings shows a return to values consistent with those at ambient conditions. However, the intra- and inter-octahedral bond angle distributions obtained for the recovered system are slightly broader than the initial ones (Figure S41), suggesting that the pressure-induced local disorder is not entirely eliminated after pressure release. This is underscored by the inability of RMCProfile to perfectly fit the XRD data, as demonstrated in Figure S42. The ability to fit the data after pressure release backtracking from the 5 GPa data demonstrates that the model is capable of accurately capturing the reversibility of the structural transition, effectively simulating the recovery process from a partially amorphous phase back to a more crystalline state. This reversibility is likely due to the partial preservation of the Pb sublattice under pressure. While Cs cations and Br anions become increasingly displaced, probably because of weaker bonding for the former and higher polarizability for the latter, the Pb atoms retain a degree of spatial coherence through their extended connectivity, allowing the structure to recover upon decompression.

Our results show that local disorder increases significantly within the orthorhombic phase of CsPbBr3 while the average crystallographic symmetry remains preserved. This allows us to define the extent of structural distortion that the phase can accommodate before undergoing amorphization. We identify the progressive development of non-uniform octahedral tilting and A-site displacement as key features leading to the loss of long-range order. Crucially, we also track the reversibility of these changes upon decompression, showing that the PbBr6 connectivity is largely retained, which facilitates structural recovery. Understanding where and how amorphization originates helps identify the structural weaknesses of the phase, providing a basis for designing materials that are less prone to amorphization under external stimuli. Moreover, our findings support the idea that structural disorder, when carefully characterized and controlled, can be used as a design element in materials synthesis, enabling the stabilization of new phases and the tuning of functional properties through compositional modification, strain, or dimensional control44.

This study highlights the capability of combining total X-ray scattering with RMC big-box modeling to uncover the complex interplay between local structural distortions and long-range crystalline behavior in CsPbBr3, a prototype three-dimensional halide perovskite, on compression. Our approach reveals that pressure-induced distortions in the PbBr6 octahedra and Cs coordination environment precede the onset of the amorphization at 2 GPa, providing critical insights into structural rearrangements that are obscured in traditional XRD. By examining both short-range and long-range structural correlations through PDF analysis, we elucidate how local disorder impacts optical properties within the crystalline phase and gain information about the mechanisms underlying pressure-induced amorphization and reversibility upon pressure release. Specifically, the non-monotonic bandgap evolution observed below 2 GPa, changing from a red to a blue shift, is directly linked to the interplay between bond compression and octahedral tilting, rather than to a phase transition as previously believed. Beyond 2 GPa, while the Pb sublattice retains partial order, significant disorder emerges, particularly in the Cs and Br coordination environments, resulting in a partially amorphous structure. Upon pressure release, the recovery of the crystalline phase is facilitated by the stability of the Pb sublattice, although residual strain and subtle local distortions persist. The reversible amorphization highlights the dynamic interplay between long-range and local structural behavior under pressure. These findings underscore the value of large-box modeling as a powerful tool to probe the structural drivers of both the crystalline and amorphous phases, enhancing our understanding of pressure-induced phenomena in halide perovskites. While this study focuses on an all-inorganic perovskite, the same approach can be extended to hybrid halide perovskites, particularly given recent advances in RMCProfile that enable rigid-body modeling of molecular cations. This integrated approach supports the design of materials with tailored properties for optoelectronic applications under extreme conditions.

Methods
Samples synthesis and characterization
All chemicals were used as received from commercial vendors. To synthesize CsPbBr3 powder, 400 mg of PbBr2 and 232 mg of CsBr were dissolved in 4 mL and 12 mL of dry dimethyl sulfoxide, respectively. The two solutions were combined and then added to 100 mL of dry acetonitrile. An orange powder slowly precipitated after 5 minutes of stirring. The CsPbBr3 powder was isolated by filtration and dried under reduced pressure. A milling jar was filled with toluene, 360 mg of the perovskite powder, and zirconia balls (ca. 5 g). The sample was ball milled at 500 rpm with 8 cycles of 30 minutes of milling and 30 minutes of rest. The sample was further dried under reduced pressure and stored in desiccant to avoid decomposition from moisture exposure. The crystal structure of CsPbBr3 was verified by both laboratory and synchrotron powder XRD at ambient conditions. A high-resolution powder XRD diffractogram of the sample was collected in a capillary at the 11-ID-B beamline of the Advanced Photon Source and is shown in Figure S1 together with the corresponding Rietveld refinement.

High-pressure total X-ray scattering measurements
Ball-milled CsPbBr3 powder was loaded into a sample chamber drilled in a pre-indented stainless-steel gasket, which was subsequently compressed between a pair of diamond anvils with 300-μm culets in a short symmetric DAC. A low-viscosity poly(dimethylsiloxane) oil (CAS 63148-62-9, Thermo Fisher; 45–55 mPa·s at 25°C) was used as the pressure-transmitting medium. The diamond-seat-cell combination has a large aperture angle of ~90° to access a large Q range. Pressure was measured within 0.1 GPa using the ruby fluorescence method45. High-pressure total X-ray scattering data were collected at beamline 11-ID-B of the Advanced Photon Source at Argonne National Laboratory using an X-ray energy of 86.7 keV (0.143 Å)46. The 2D scattering patterns were masked to remove diamond reflections and integrated using Fit2d software47. Scattering data collected using the empty DAC, with the gasket pre-indented and drilled, was used for background subtraction after the 2D integration when extracting the PDFs46. The PDFgetX2 software program was used to extract the structure factor S(Q) up to Q=18 Å−1 and to transform S(Q) to the real-space PDF G(r) for initial analysis48. Subsequent analysis of the PDFs using RMC simulations involved transforming S(Q) internally, normalizing to an absolute scale using StoG within the RMCProfile program21. A Fourier filter with low-r cutoff at 2.64 Å and Lorch window function were used in the Fourier transform. The maximum value for Q for the Fourier transform was set at 17.58 Å−1. A detailed example of this processing workflow is presented in Figure S5, showing the extraction of G(r) from the raw data collected at 0.0 GPa in the DAC. For use in RMCProfile, G(r) was normalized using StoG to match the definition that G(r)=–1 at low-r, consistent with the formalism described by Keen49. We also collected powder XRD data at a longer sample-to-detector distance for average structure analysis. Rietveld (0–1.6 GPa) and Le Bail (2–5 GPa) refinements of the high-pressure data were performed using GSAS-II, while FullProf was used for the Rietveld analysis of the capillary diffractogram and for the comparison between the starting and recovered sample measured in the DAC.

RMC simulations
RMC simulations were conducted using RMCProfile (version 6.7.9.5) to analyze the experimental data21. A 6×6×4 supercell containing 2880 atoms was constructed, with initial atomic positions derived from the experimental crystallographic structure obtained through Rietveld refinement. The simulation aimed to fit the experimental real-space PDF, particularly focusing on the G(r) function which is defined as:

G(r) = 1/(2π)3ρ0 ∫ 0∞ Q2F(Q) sin(Qr)/Qr dQ (1)

Where F(Q) is the total scattering structure factor, given by:

F(Q) = ρ0 ∫ 0∞ 4πr2G(r) sin(Qr)/Qr dr (2)

Here, ρ0 = N/V represents the average atomic number density, expressed in atoms per Å3. The G(r) represents the probability of finding a pair of atoms separated by a specific distance in real space, providing information on local density variations due to the arrangement of atoms. This method helps in understanding the local atomic arrangements beyond what is observable from Bragg peaks alone, including distortions and short-range order in the material. For a more detailed explanation of the formalism, we refer to the papers by Keen49 and Peterson et al.17.

To prevent unphysical atomic configurations, several approaches have been used. In an initial test at low pressures, polyhedral restraints were applied to maintain the integrity of PbBr6 octahedra. This restraint, implemented following the RMCProfile formalism, adds a χ2 penalty term based on deviations from ideal bond lengths and angles within the octahedra. The ideal geometry was defined from the ambient-pressure configuration and provided as input for the simulation. The restraint acts to preserve the topology of the octahedra without dominating the experimental data, with weighting factors being set low enough to ensure that the final polyhedral geometry is determined by the data rather than by the restraint21. In addition, a “distance-window” restraint was applied to define minimum and maximum allowed distances rij between a pair of neighboring atoms, where i and j denote the species of atoms in the pair. Finally, a bond valence sum restraint was employed on the whole data set to limit bond distances and further ensure chemically reasonable configurations. RMCProfile was also used to fit the Bragg data. The RMCProfile simulations were run for ~12–30 h each, which correspond to 1 to 3 million accepted moves. An example of the evolution of χ2 is shown in Figure S43.

High-pressure visible absorption measurements
Optical absorption measurements were collected at beamline 22-IR-1 of the National Synchrotron Light Source II at Brookhaven National Laboratory. Visible absorption measurements between 400 and 1000 nm were performed using a customized visible microscope system equipped with an IsoPlane SCT-320 Imaging Spectrograph, a PyLon CCD detector (Princeton Instruments) and a tungsten light source. A symmetric DAC with type IIas diamond anvils of 500-μm culets was used for the absorption measurements. The sample chamber, a 200-μm hole drilled in a pre-indented stainless-steel gasket, was filled with KBr as the pressure-transmitting medium50. A thin pellet of the sample, a few microns thick, was placed to occupy half of the chamber area. At each pressure point, a reference transmission spectrum was collected through KBr before measuring the sample transmission under the same beam conditions (e.g., energy range and aperture size). The spectra were analyzed using the Tauc-plot method to determine the bandgap evolution as a function of pressure.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Celeste and co-workers describes a pair distribution function analysis of pressure-induced amorphisation in the well-known lead halide perovskite CsPbI3. This technique is uniquely sensitive to local structure, in contrast with the space average described by conventional crystallographic methods. It thus offers important insight into this material's pressure response - both (directly) the change in structure, and (indirectly) the change in optoelectronic properties such as its increased band gap. The authors use the RMCprofile program to perform \"big-box\" analysis of their data, cleverly adapting this method, which usually relies on a prior Rietveld refinement, to be appropriate even at pressures when significant partial amorphisation is observed. They conclude that the putative isostructural phase transition at about 1.6 GPa is instead the result of cumulative disorder, and suggest that the reversibility of the amorphisation process may at least partially be attributed to the fact that the Pb sublattice remains relatively intact.\nThis is a technically sound paper, and collecting PDF data of this quality from a pressure cell is itself an impressive experimental achievement. The data are comprehensively analysed and the manuscript is well presented, including for readers who may not be experts in this specific technique or material. An extremely comprehensive set of SI will, among other things, allow the modelling process to be rigorously reproduced. My comments are, therefore, relatively minor.\nFig. 1b: The PDFs should be presented down to r = 0 rather than truncated, so their low-r behaviour, which is an important guide to the quality of the data processing, can be evaluated.\n\"the 3.5 and 5.5 A range\": should \"and\" read \"to\" here? More importantly it is difficult to see exactly what the authors are referring to here: perhaps the relevant section could be highlighted or inset in Fig. 1b.\nFig. 5: could this be displayed in terms of the aristotype cubic perovskite structure rather than the orthorhombic cell? (Or, easier, could the crystallographic structure be given as a separate part of the figure, for ease of comparison?)\nHave the authors considered analysing and/or presenting this in terms of \"ADP\" ellipsoids? In particular the distribution of Br atoms seems interesting, in that it appears to be elongated in a particular direction, but this is very difficult to read precisely from the figure as it stands.\nConclusion, l.303: In principle PDF methods can indeed be used to study dynamics, but it doesn't seem that this was the case here; the authors might consider rephrasing.\nCould any more be said about the curious variation in lattice parameters with pressure, Fig. S3? In particular, how does the model of increased amorphisation above 2 GPa explain noticeable features in a and b but not c, which indeed resemble a phase transition?\nTable S1 should report esds.\nIn Fig S7, which PDFgui parameters were allowed to refine?\nWhich version of RMCprofile was used? There are significant differences between versions 6 and 7.\nA very minor issue, which I mention only because spell check won't pick it up: l.204 should probably not refer to \"bong lengths\".", "reviewer_2": "Celeste et al. present a very interesting manuscript about the total scattering upon compression of CsPbBr3. The material studied is for sure relevant, due to the large range of possible applications and its local structure is relatively less explored with respect to other perovskite and perovskite-like materials. The methodology employed, PDF studies at high pressure, is for sure challenging and the possibility of combining a local structure probe with high pressure intriguing.\nIn my opinion, the interest and novelty of the topic are well in line with the scope of the journal. However, before publication, the manuscript needs to be revised. In particular, the major flaw that I find in the text is the description of the methodology, both in the experiment itself and in the data analysis. At the present state, the research described here seems hard to be reproduced.\nThe manuscript would also benefit from a more careful description of the starting materials and some comparison with the local structure of this and similar materials, at least at low temperature.\nHere I attach some more detailed comments.\nLine 60: structural models which describe instead of describes\nLine 75-78: “an archetypal three-dimensional (3D) halide perovskite. Indeed, the simplest halide perovskite structure is the 3D perovskite”, : I know that, especially in the material science community, perovskites and perovskite-like structures are described in terms of the so-called dimensionality, but there is really no need to use such a jargon, which is most of the time wrong from the crystallographic point of view. Additionally, I find this use of the adjective “archetypal” together with the reference to “the simplest perovskite structure is the 3D perovskite” misleading. “The 3D perovskite structure” does not exist, all crystals showing one of the 230 space groups are 3D. The aristotype perovskite structure, which is cubic space group Pm-3m, does exist and is usually considered the ideal perovskite structure. The terms reported in the text could hint that the material crystallizes in the aristotype structure, whereas at room temperature CsPbBr3 is usually reported as orthorhombic. I would modify the sentence in something like “shows a distorted perovskite structure, space group Pnma or Pbnm”. Ironically, this is the space group in which the mineral perovskite actually crystallizes, but from the context it does not look like that the authors are referring to this when the write “archetypal” or “the simplest structure”. However, recent studies [Khan, Tahira, et al. \"Mapping and Characterization of Local Structures of CsPbBr3.\" ACS omega 9.33 (2024): 35789-35797.; Liu, Sizhan, et al. \"High‐Resolution In‐Situ Synchrotron X‐Ray Studies of Inorganic Perovskite CsPbBr3: New Symmetry Assignments and Structural Phase Transitions.\" Advanced Science 8.18 (2021): 2003046.] have suggested that CsPbBr3 could have a monoclinic superstructure. The monoclinic and orthorhombic superstructure might be indistinguishable in the powder patters, but this aspect should discussed in the text. I would also remove or rephrase the sentence “CsPbBr3 exemplifies this structure”, because the topics discussed above.\nLine 99: the use of silicone oil as a pressure transmitting medium always makes the scenario a bit tricky because of the large amount of different compounds under this name and its low hydrostatic limit. I think it could be useful to add in the supplementary information which compound was actually use to enhance the reproducibility of the results.\nLine 141: I think it is useful to clarify here or in the methodology section which exactly is the polyhedral restraint.\nFigure 4: I do not understand why panel b is so much smoother than the others and the PDF at 0 and 1.6 GPa seems to have a strange trend, with the peaks changing abruptly between 3.25 and 3.5 Å\nLine 251: Have the authors calculated the distortion parameters (Robinson et al. 1971) for the octahedra? Can the intra-octahedral distortion be related to the optical properties?\nLine 259: The figure cited here is S26, but there are two S26 figures in the supplementary, so figure S26 at line 277 should be S27 and all the other references to the figure in the supplementary information after line 259 should be checked and corrected.\nMethodology\nThe software or codes used to perform the Rietveld or LeBail refinements is never stated it the text, it would be useful to add this information.\nI think it could be interesting to add in the supplementary information one or more graphs illustrating the extraction of the PDF, including the background.\nThe Rwp of the refinements are never reported.\nSupplementary information\nTable S1 there are no uncertainties reported on the unit cell parameters\nFigure S3 Due to the huge difference between c and the other two axes, it could be clearer to report a and b in a different graph, or to insert a break along the y axis.\nThe label S26 is used twice, so the numbering should be amended.", "reviewer_3": "The manuscript by Celeste et al. presents a detailed study that combines high-pressure total X-ray scattering with reverse Monte Carlo big-box modeling to investigate local structural evolution and partial amorphization in CsPbBr3. The authors reveal the role of local disordering of Cs and Br, while the stability of the Pb sublattice, in enabling reversible pressure-induced amorphization. The experimental design is rigorous, and the modeling approach is appropriately applied, yielding well-supported and technically sound results.\nThat said, while the study provides valuable insights into the local structural distortion of halide perovskites under pressure, I am not convinced that its significance in its current form reaches the threshold required for publication in Nature Communications. In particular, the broader implications for the field of 3D halide perovskites are somewhat underdeveloped. For instance, the abstract states that “unraveling the short-range mechanisms behind this reversibility could provide key insights into phase stability and disorder recovery, paving the way for new strategies to stabilize metastable phases in halide perovskites.” However, this forward-looking perspective is not sufficiently expanded upon in the main text. The discussion would benefit from a clearer articulation of how the reported findings might guide future materials design strategies or influence the development of pressure-responsive perovskite materials.\nIn addition, a more thorough discussion of how pressure-induced local structural responses—beyond the observed bandgap evolution—might impact other optoelectronic properties would strengthen the manuscript’s relevance to the broader perovskite research community.\nMinor Comments:\n1. The correlation between bandgap evolution and local structural distortions is important to support the conclusions. While the pressure-dependent optical spectra are shown in Figure S23, I recommend including a figure plotting the extracted bandgap values alongside key structural parameters to make this relationship more accessible to readers.\n2. The reversibility of amorphization is attributed to the preservation of the Pb framework. Can the authors elaborate on the physical origin of this robustness relative to Cs or Br?\n3. Line 204: “bong lengths” should be corrected to “bond lengths.”", "reviewer_4": "Celeste et al. investigated pressure-induced structural changes and amorphization in CsPbBr3 using total X-ray scattering and Reverse Monte Carlo modeling. Local distortions, including PbBr6 octahedral tilting and cesium displacement, appear below 2 GPa, influencing optical properties such as bandgap evolution. Between 2 and 5 GPa, partial amorphization occurs, characterized by increased lattice disorder, while the Pb sublattice remains intact. Upon releasing pressure, the material largely reverts to its original crystalline state, although residual strain and subtle distortions persist. The research demonstrates the dynamic interplay between local and long-range structural behavior, elucidating how disorder governs the pressure-induced structural behavior of halide perovskites. I recommend the publication of this work in Nature Communications after addressing the following comments:\n1. The authors focus on the mechanisms underlying reversible pressure-induced amorphization in CsPbBr3. The results are interesting and significant, yet more discussion is suggested to emphasize the impact of this work. Specifically, highlighting the mechanistic insights gained from the experimental and theoretical approaches used would further strengthen the importance of its contribution to the field.\n2. While Reverse Monte Carlo \"big-box\" modeling is employed to investigate local distortions, there is no discussion of its universality. How does this method perform when applied to more complex systems, such as organic-inorganic hybrid halide perovskites?\n3. On page 4, the authors state that “Beyond 2 GPa, the polyhedral constraint failed, suggesting bond breaking as pressure increased”. Are the authors specifically referring to the breaking of Pb-Br bonds? This requires a more detailed and convincing explanation.\n4. On page 8, the authors claim that the displacement of Cs cations from their ideal positions results in an overall loss of crystallinity. Nevertheless, Cs atoms typically possess the freedom to move within the perovskite cages. The crystallinity of the lattice mainly hinges on the integrity of the Pb-Br framework.\n5. In Figure 4, it would be advantageous to include a pressure point ranging from 0 to 1.6 GPa. Such an addition would aid in the comprehension of the structural evolution before and after amorphization.\n6. High-pressure single-crystal XRD studies often report the emergence of grain boundaries and defects within a crystal, which may be related to pressure-induced amorphization. More in-depth discussion on this aspect is suggested."}, "author_response": "We thank the reviewer for the positive evaluation of our manuscript and greatly appreciate the recognition of the data quality, our methodological approach, and the insights gained into the pressure-induced structural evolution of CsPbBr . We have carefully addressed all the reviewer’s comments in the revised manuscript and provided a detailed point-by-point reply below.\nFig. 1b: The PDFs should be presented down to r = 0 rather than truncated, so their low-r behaviour, which is an important guide to the quality of the data processing, can be evaluated.\nWe agree that presenting the PDFs over their full range down to r = 0 helps assess the quality of the data processing. We have updated panel b in Figure 1 to include the low-r region, and the revised figure is included in the new version of the manuscript.\n\"the 3.5 and 5.5 A range\": should \"and\" read \"to\" here? More importantly it is difficult to see exactly what the authors are referring to here: perhaps the relevant section could be highlighted or inset in Fig. 1b.\nWe thank the reviewer for catching this typo, which has been corrected as suggested. The sentence now reads: “On increasing pressure, we observe short-range structural modifications even below 1.6 GPa, particularly in the 3.5 to 5.5 Å range (Figure 1c), where primary contributions arise from the Br-Cs, Br-Br, and Pb-Cs first-neighbor distances.”\nWe have also added a new panel in Figure 1, namely panel (c), to highlight the evolution of the PDFs up to 2 GPa in the 3.3-5.5 Å range. The figure’s legend has been updated accordingly and now states: “Figure 1: Experimental powder XRD patterns (a) and X-ray PDF data (b, c) of CsPbBr under pressure. Panel (c) shows a zoomed-in view of the G(r) up to 2 GPa in the 3.3–5.5 Å range, highlighting the evolution of features in this r range. Patterns are stacked vertically for clarity. Patterns collected after the pressure was released are shown in orange.”\nFig. 5: could this be displayed in terms of the aristotype cubic perovskite structure rather than the orthorhombic cell? (Or, easier, could the crystallographic structure be given as a separate part of the figure, for ease of comparison?)\nSince CsPbBr does not crystallize in a cubic perovskite structure at room temperature, we believe that displaying Figure 5 using this symmetry would be misleading. CsPbBr is commonly described as orthorhombic (space group Pbnm), although, as noted by Reviewer 2, recent studies have suggested the presence of monoclinic superstructures that may not be distinguishable in powder diffraction data [ACS Omega 2024, 9, 33, 35789; Adv. Sci. 2021, 8, 18, 2003046]. We agree that providing a reference structure under ambient conditions can aid in interpreting Figure 5 and help contextualize the atomic displacements shown. For this reason, we have added both a representation of the ideal orthorhombic structure viewed along the three crystallographic axes (Figure S3) and a visualization of the ideal supercell configuration prior to RMC refinement (additional panel in Figure S30) to the Supplementary Information. These serve as structural references to better appreciate the effects of pressure on the atomic configurations.\nHave the authors considered analysing and/or presenting this in terms of \"ADP\" ellipsoids? In particular the distribution of Br atoms seems interesting, in that it appears to be elongated in a particular direction, but this is very difficult to read precisely from the figure as it stands.\nFollowing the reviewer’s suggestion, we performed an anisotropic displacement parameters (ADPs)-like analysis of the positional distribution of each atomic species, with a focus on Br atoms. While RMCProfile can compute ADPs relative to the initial positions in the primitive cell, we developed a Python script to statistically analyze the atomic distributions from the RMC supercell. Atoms were grouped according to their symmetry-equivalent positions in the crystallographic reference structure, resulting in 4 Cs, 4 Pb, 4 Br1, and 8 Br2 atoms per unit cell. This approach is better suited to systems under pressure or with significant disorder and provides clearer insight into anisotropic displacement trends.\nIn the Pbnm symmetry, CsPbBr features two distinct types of Br atoms: Br1, which occupies the 4c Wyckoff position (x, y, ¼) and forms Pb–Br1–Pb chains lying in the a–c plane, and Br2, which occupies the 8d position (x, y, z) and forms Pb–Br2–Pb chains that are tilted out of the a–b plane due to the octahedral rotations (see Figure S4) [ACS Omega 2020, 5, 11, 5931; Chem. Commun. 2024, 60, 14200]. For each symmetry-equivalent site, we computed the displacement of every atom from its ideal reference position along the crystallographic a, b and c axes. We then formed the 3x3 covariance matrix of these displacement vectors and extracted its diagonal elements σ , σ and σ , representing the statistical analogue of the crystallographic ADP parameters U , U and U . By diagonalizing the same covariance matrix, we obtained the eigenvectors and eigenvalues, which we used to plot 3D ellipsoids overlaid on the ideal Cs, Pb, and Br positions in the unit cell.\nA representation of 12 ellipsoids for each Br site is now shown along the a-, b- and c-axis view at selected pressures (0.0, 0.8, 2.0 and 5.0 GPa) in the updated Supplementary Information (Figures S31-S34). Ellipsoids associated to Br1 and Br2 atoms are shown in orange and blue, respectively, to highlight their different displacement behaviors.\nTo quantify the directionality and anisotropy of Br atom displacements, we plotted the average σ values for each site group as a function of pressure and calculated the anisotropy as the ratio between the minimum and maximum eigenvalues (with values closer to 1 indicating lower anisotropy) [Acta Cryst., 1996, A52, 770-781]. The results are reported below and in Figures S35-36. At 0.0 GPa, Br1 atoms exhibit moderate anisotropy ( 0.7), with ellipsoids elongated primarily along the b-axis (σ > σ > σ ) while Br2 atoms show stronger anisotropy ( 0.45) and more directionally constrained ellipsoids, with σ nearly double than σ and σ , indicating elongation mainly along the c-axis. These differences align with the crystallographic roles of the two Br sites in the Pbnm structure: Br1 atoms, confined near the a–c plane (Wyckoff position 4c, z = ¼), are more susceptible to collective tilting of the octahedra, especially involving displacements along b, while Br2 atoms (8d position), which lie off the mirror plane and participate in Pb–Br2–Pb connections tilted out of the a–b plane, are more directly affected by distortions perpendicular to that plane. These anisotropic characteristics reflect symmetry-imposed degrees of freedom and show how local octahedral dynamics manifest differently for axial (Br1) and equatorial (Br2) Br atoms.\nUpon increasing pressure, we observe a sharp increase in all σ parameters for both Br sites around 2.0 GPa, consistent with the transition to a more disordered, partially amorphous phase. Beyond this pressure, Br1 atoms primarily elongate withing the a-b plane, while Br2 atoms retain dominant displacement along the a-axis. However, the directional preferences diminish, as the anisotropy decreases for both sites, more noticeably for Br2 (see Figure S36b), indicating more isotropic, rounded ellipsoids. This evolution suggests that pressure reduces the directional coherence governed by octahedral tilting, promoting local configurational disorder as the dominant source of atomic displacement. Interestingly, both Br1 and Br2 show enhanced anisotropy near the transition region (around 1.6-2.0 GPa), suggesting that directional displacements may become temporarily more pronounced just before the structure enters a more disordered phase.\nConclusion, l.303: In principle PDF methods can indeed be used to study dynamics, but it doesn't seem that this was the case here; the authors might consider rephrasing.\nWe agree that the term “dynamics” may be misleading in the context of our study. We have therefore revised the sentence in the Conclusion to clarify this point. The updated sentence at line 303 now reads: “Our approach reveals that pressure-induced distortions in the PbBr octahedra and Cs coordination environment precede the onset of the amorphization at 2 GPa, providing critical insights into structural rearrangements that are masked in traditional XRD.”\nFollowing the same advice, we have also rephrased our final sentence of the Introduction by replacing “dynamics” with “configurations”. The revised sentence is the following: “Our study highlights the critical importance of understanding both the local and average structural configurations in halide perovskites to get a more accurate picture of the structure–property relationships.”\nCould any more be said about the curious variation in lattice parameters with pressure, Fig. S3? In particular, how does the model of increased amorphisation above 2 GPa explain noticeable features in a and b but not c, which indeed resemble a phase transition?\nBefore addressing the reviewer’s comment on the variation of lattice parameters with pressure, we would like to mention that we have taken steps to improve the clarity of Figure S3 (now Figure S6 in the revised Supplementary Information), following a suggestion from Reviewer 2. Specifically, we have introduced a break on the y-axis, which we believe significantly improves the readability and allows for better visualization of the pressure-dependent trends of the lattice parameters a, b and c. In the updated Figure S6, also shown below as Figure R3, a subtle slope change in the pressure dependence of the c parameter becomes more apparent, whereas it was less noticeable in the original plot. We have also added the volume as a function of pressure, along with a fit to a second-order Birch–Murnaghan (BM) equation of state (EoS) up to 1.6 GPa (before the onset of amorphization), which yields a bulk modulus of 20 GPa. Above this pressure, the extracted volumes refer only to the residual crystalline fraction and do not represent the full system. Therefore, we do not extend the EoS fit beyond 1.6 GPa and have added a note in the figure caption to clarify this and avoid potential misinterpretation.\nAs the reviewer points out, the lattice parameters exhibit distinct responses to pressure: a and b show noticeable features around 2 GPa, while the evolution of c appears smoother. We interpret this anisotropic behavior in the context of pressure-induced amorphization, which interacts differently with the three crystallographic directions due to the underlying orthorhombic Pbnm structure of CsPbBr . In this phase, octahedral tilting involves out-of-phase rotations along the a and b axes and in-phase tilting along c (Glazer tilt system a-a-c+), making a and b more sensitive to local disorder. In contrast, the c-axis features corner-sharing Pb–Br–Pb chains that remain more aligned due to the in-phase nature of the tilting, offering greater resistance to distortion [PRM 2018, 2, 025401].\nAs amorphization begins around 2 GPa, local disorder becomes the dominant feature, overtaking octahedral tilting. This results in non-monotonic trends and a plateau in the a and b lattice parameters, while c continues to evolve more smoothly, possibly because it accommodates distortion earlier in the pressure range or because its structural rigidity resists further compression.\nThis interpretation is further supported by prior structural refinements [J. Phys. Chem. Lett. 2017, 8, 372], which showed stronger reductions in Pb–Br–Pb angles within the a–b plane compared to those along c up to 2 GPa. We have added an explanatory note in the SI to help clarify this point.\nTable S1 should report esds.\nTable S1 has been updated to include uncertainties (esds) alongside the refined values.\nIn Fig S7, which PDFgui parameters were allowed to refine?\nIn the PDFgui refinement, we allowed the lattice parameters (a, b, and c), the scale factor, and the correlated motion parameter delta1 to refine. Atomic positions were refined for Cs and Br1 along x and y, and for Br2 along x, y, and z. Displacement parameters U , U and U were refined for all Cs, Pb, and Br atoms, with a single value applied across the three directions for each atomic species. Constraints were applied to enforce symmetry equivalence within each atomic site. The Qdamp and Qbroad parameters were fixed at 0.0358 and 0.01946, respectively. This information has now been added to the caption of the updated Figure S10 in the revised SI.\nWhich version of RMCprofile was used? There are significant differences between versions 6 and 7.\nAll results presented in this study were obtained from simulations run using RMCProfile version 6.7.9.5. This information has been added to the Methodology section of the manuscript.\nA very minor issue, which I mention only because spell check won't pick it up: l.204 should probably not refer to \"bong lengths\".\nThe typo at line 204 has been corrected, and “bong lengths” now reads “bond lengths”."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have fully responded to my comments and, as far as I can see, the other referees'. The division of the new information between main text and supplementary information is entirely appropriate. I am pleased to recommend the manuscript for publication in its current form.", "reviewer_2": "I a reviewing this manuscript for the second time and I am glad to see that my comments, together with the suggestions of the other reviewers , have been carefully evaluated and taken into account by the authors.\nI think the manuscript has greatly improved and I am particularly happy of the nice correlation between the bandgap and the octahedral distortion. As far as I am concerned, I think that the work is now ready for publication.\nI have only a curiosity, which the authors could eventually address in the supporting material. It is now stated that the sample was ball-milled and ball-milling can produce relatively small crystallites. At least in the low pressure range, the crystal size was evaluated? What effect a small size could have on the data?", "reviewer_3": "The authors have addressed my questions. I recommend publication of this manuscript", "reviewer_4": "The authors have addressed my concerns in the latest revision, and I therefore recommend the publication of this work in Nature Communications."}, "author_response": "We thank the reviewer for their thorough evaluation and are grateful for the recommendation to publish the manuscript in its current form.\nWe appreciate the reviewer’s positive assessment and provide our responses to the questions below.\nAs mentioned, the sample was ball-milled to enhance powder averaging. While no direct crystallite size measurement was performed on this batch, similar procedures typically yield particles in the 1–5 µm range. We have added a note in the Supplementary Information stating this, along with a brief discussion of the implications. Specifically, crystallite sizes in this range are advantageous for total scattering experiments: they are large enough to produce well-defined Bragg peaks, as seen in our data, yet small enough to minimize preferred orientation and multiple scattering, which improves PDF quality. We have included a reference to Keen, Crystallogr. Rev. (2020), in support of this context.\nWe thank the reviewer for their recommendation and are glad that the revisions addressed all concerns.\nWe thank the reviewer for their support for publication."}]

Decision letter