Experimental Observation of Possible Pressure-Induced Phase Transformation in GdAlO 3 Perovskite Using In Situ X-ray Diffraction

: Gadolinium aluminate perovskite (GdAlO 3 ) was studied at high pressures of up to 23 GPa in a diamond anvil cell (DAC) using monochromatic synchrotron X-ray powder diffraction. Evidence of a pressure-induced phase transformation from orthorhombic ( Pbnm ) to rhombohedral ( R 3 c ) structure was observed at 21 GPa and further proved by DFT calculations. Before phase transition, the volumetric ratio of polyhedron A and B (i.e., V A /V B for ABX 3 general notation) in the Pbnm phase continuously increased towards the ideal value of ﬁve at the transition, indicating a pressure-induced decrease in the structural distortion as opposed to the trend in many other orthorhombic perovskites (e.g., CaSnO 3 , CaGeO 3 , MgSiO 3 and NaMgF 3 ). Pressure–volume data of the Pbnm phase were ﬁtted to the third-order Birch–Murnaghan equation of state yielding a bulk modulus ( K o ) of 216 ± 7 GPa with a pressure derivative of the bulk modulus ( K (cid:48) o ) of 5.8 GPa (ﬁxed). This work conﬁrms the pressure-induced phase transformation from orthorhombic to a higher symmetry structure previously predicted in GdAlO 3 perovskite.


Introduction
Perovskite materials are well-known solids with the general formula of ABO 3 , where A is a large cation occupying the voids with coordination number 12 (in the ideal cubic perovskite structure) and B is a smaller cation in octahedral oxygen coordination.Rare-earth aluminates with perovskite structure are potential materials for optical, electronic, structural, and magnetic applications such as colossal magneto resistant materials, substrates for high-temperature superconductors, solid oxide fuel cells and microwave dielectric materials [1][2][3][4][5][6][7].A large number of rare-earth perovskite-type oxides have been studied because of their interesting properties.Many of those ABO 3 compounds crystallize into an orthorhombic structure which deviates from the ideal cubic structure via the titling and distortion of the BO 6 octahedra [8,9].The distortion degree of perovskites can be described with the following parameters: the BO 6 octahedral tilting and rotation, the Goldschmidt's tolerance factor(t) and the polyhedron volume ratio V A /V B , where V A is the volume of A-site polyhedron and V B is that of the B-site octahedron.The values for the tilting angle ( B-O1-B), the rotation angle ( B-O2-B), t and V A /V B for an ideal perovskite are 180 • , 180 • , 1 and 5, respectively.The nature of this distortion has been related to the existence of different ionic radii within the crystal [10][11][12][13][14][15][16].Pressure, as one of the important thermodynamic parameters plays a critical role in altering material properties.Under pressure, shortening of the effective ionic radius may vary depending on the ion itself and its coordinating environment.Zhao et al. [17] developed a model using the bond valence concept to predict the structural behavior of oxide perovskites at high pressure, introducing the ratio M A /M B , which determines the relative compressibility of the AO 12 and BO 6 sites due to the change in the average bond distance, where M A and M B describe the variation of the bond valence sum at the central cation in a polyhedral site.Based on this model, the relative polyhedral compressibility (i.e., V A /V B ) determines the influence of pressure on the crystal structure of perovskite [18].Some orthorhombic rare-earth perovskites such as LaFeO 3 and PrFeO 3 undergo a first-order transition to high-pressure phases, among which the high-pressure phase of LaFeO 3 was reported to have a tetragonal symmetry with unknown structural details [19].First-order reversible structural phase transition from the orthorhombic to rhombohedral R3c structure was observed in SmAlO 3 and LaGaO 3 near 10 GPa [20] and 2.5 GPa [21], respectively, by X-ray synchrotron powder diffraction.
GdAlO 3 orthorhombic perovskite has been studied as a potential material for industrial applications considering its magnetic, optical, electronic, and structural properties [22][23][24][25][26][27].The crystal structure of gadolinium aluminate has been reported at ambient conditions [22,28,29].Ross et al. [30,31] studied the structural behavior of GdAlO 3 at high pressures of up to 8 GPa and found that the abovementioned structural distortion decreases with increasing pressure.By extrapolating the metrical deviation of the orthorhombic perovskite from tetragonal symmetry (proportional to (b−a)/a 0 where a 0 is the value of the phase with a higher symmetry), they predicted an orthorhombic to higher-symmetry transition at about 12 GPa in GdAlO 3 .Nevertheless, such a transition has never been directly observed in any experimental study.In this work, we compressed GdAlO 3 up to 23 GPa and observed signs of the predicted transition but at a higher pressure (21 GPa).Density functional theory (DFT) calculations indicate that the rhombohedral structure (R3c) found in LaGaO 3 is favorable at high pressure over the low-pressure orthorhombic structure (Pbnm) of GdAlO 3 and the tetragonal structure (I4/mcm) found in CeAlO 3 [32].

Materials and Methods
The GdAlO 3 sample was prepared through the solid-state reaction of Gd 2 O 3 and Al 2 O 3 by arc-melting in argon atmosphere.Details of the procedure can be found elsewhere [9,32,33].High pressures were generated using a Mao-Bell diamond anvil cell (DAC).Two opposed diamond anvils with 300 µm diameter culets were compressed on a 302 stainless steel gasket with an initial thickness of 250 µm.The gasket was pre-indented with the anvils to a thickness of 60 µm and a hole of a 125 µm diameter was drilled at the center of the indentation to hold the sample.The synthesized sample was grounded into fine powder in an agate mortar and loaded into the 125 µm diameter sample chamber.A methanol-ethanol (4:1) mixture was loaded together with the sample and a small ruby sphere (pressure calibrant) as a pressure transmitting medium.The in situ high-pressure powder X-ray diffraction experiments were performed at the 16-IDB beamline of HPCAT (High Pressure Collaborative Access Team) at Advanced Photon Source (APS) of Argonne National Laboratory, Lemont, IL, USA.The monochromatic X-ray beam of a 0.3757 Å wavelength was focused to a spot size of 5 µm.Diffraction patterns were recorded using a MAR345 detector.During the experiment, the sample pressure was increased stepwise; fluorescence of the ruby sphere and sample diffraction were collected at each step.The sample pressure was determined using ruby fluorescence pressure scale [34,35].The FIT2D software [36] was used to integrate the recorded 2-dimensional diffraction patterns to obtain intensity versus 2θ plots.Rietveld refinements were performed for both the ambientpressure phase and the high-pressure phase at 21 GPa using the GSAS software with a GUI interface [37,38].The volumetric compression data of the ambient-pressure phase were derived through full-spectrum Le Bail analysis of the in-situ X-ray diffraction patterns.Density functional theory (DFT) calculations were performed using the Vienna ab initio simulation package (VASP 6.3) [39,40].The generalized gradient approximation (GGA) [41] with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [42] was used to describe exchange and correlation effects.Electron wave functions were expanded by a plane wave with a cutoff energy of 520 eV.The Gaussian smearing method was used with a width of 0.05 eV.A Monkhorst-Pack [43] k-point grid was set with a 0.03 1/Å k-point separation which provides a mesh density of >1000 per reciprocal atom.Atomic relaxation was performed until the changes in the electronic and ionic steps were less than 10 −6 eV and 10 −5 eV, respectively.

Results
A sequence of in situ X-ray diffraction patterns of the sample was taken up to 23 GPa (Figure 1).All the diffraction peaks in the initial diffraction pattern were identified as the Pbnm perovskite phase in agreement with previous works [9,29,31,44].Disappearance of (111), (021), ( 113), (023)/(221), and (133) peaks indicates a possible pressure-induced structural phase transition at a pressure between 16.2 GPa and 21 GPa, and this transition is reversible as all these disappeared peaks reappear when the pressure is released back to ambient.From the symmetry relationship point of view, there are three higher symmetry phases for perovskite in the pathway from its Pbnm orthorhombic to Pm3m cubic polymorphs, i.e., P4/mbm, I4/mcm and R3c [16].However, differences in symmetry operations require that the (111) and (021) peaks in the Pbnm orthorhombic phase disappear in the I4/mcm tetragonal phase (i.e., hkl: h + k + l = 2n and 0kl: k, l = 2n) and the R3c rhombohedral phase while they are allowed in the P4/mbm tetragonal phase.Since the (111) and (021) peaks in the Pbnm orthorhombic phase completely disappeared when sample pressure increased to 21 GPa (Figure 1), the P4/mbm tetragonal phase was not considered.Therefore, we examined three possible structural models using the diffraction pattern at 21 GPa, i.e., Pbnm (the ambient pressure phase), R3c (the high-pressure phase observed in SmAlO 3 [20] and I4/mcm (the structure of CeAlO 3 ).As shown in Figure 2 and Table 1, the differences in structure refinements using the three different space groups are not significant enough to conclusively determine the favorable structure for the high-pressure phase.We then calculated the total energy of the three structure models as a function of pressure using DFT (Figure 3).The calculation results indicate that Pbnm is favorable at low pressures, whereas R3c is favorable at high pressures, although the calculated transition pressure (75 GPa) is much higher than the experimental pressure where the disappearance of diffraction peaks was observed.We realize that the methanol-ethanol (4:1) pressure medium solidifies at a pressure above 10 GPa [45], creating a non-hydrostatic pressure environment and therefore the sample may experience significant stresses at 21 GPa.Zhao et al. [46,47] were able to quantify the stress field using single-crystal X-ray diffraction and indicate that such a non-hydrostatic pressure environment in a DAC may affect and even induce the phase transition as stress fields can influence the structural distortions in perovskites.A more recent experimental study [48] demonstrated that applying non-hydrostatic pressure may lower the transition pressure in a DAC.Taking this stress influence and possibly large uncertainty in determining transition pressure using DFT calculations into consideration, we propose that the theoretically predicted Pbnm to R3c phase transition may occur at a pressure between 16.2 GPa and 21 GPa promoted by non-hydrostatic pressure.As shown in Table 1, the volume per formula unit of the R3c structure is nearly identical to the Pbnm structure within the experimental uncertainty.To further confirm the symmetry of the high-pressure phase, higher resolution structure characterization, e.g., in situ single crystal X-ray diffraction, is needed.The structural deviation of the perovskite (ABX3) structure from its ideal cubic structure is commonly characterized by its BX6 octahedral titling.The Pbnm perovskite has a simple tilt system of a − a − c + [49,50] (the superscript + and -indicate that successive octahedra along an axis have the same tilt and opposite tilt respectively).Figure 3a shows the GdAlO3 structure model based on the refinement of the X-ray diffraction at ambient condition.The tilting angles along the a-axis and the b-axis are identical, and successive octahedra along each of these axes have opposite tilt, whereas successive octahedra along the c-axis have the same tilt.The R3 c perovskite has a tilt system of a − a − a − [49,50].Figure 3b shows the high-pressure phase structure model of GdAlO3 based on the refinement of the X-ray diffraction at 21 GPa.There are three identical tilts along the a, b and c-axis, and successive octahedra along each of the axes have opposite tilt.The degree of structure distortion of perovskite can be evaluated using the volumetric ratio (VA/VB) between AX12 and BX6 polyhedron with an ideal value of five for undistorted cubic structure [15].Assuming regular BX6 octahedra throughout the Pbnm perovskite structure, VA and VB can be calculated through its unit cell dimensions [15,51].Evolution of the VA/VB ratio of the low-pressure phase in GdAlO3 with pressure is given in Figure 4 shows the VA/VB ratio as a function of pressure (Table 2).As pressure increases, the VA/VB ratio of the Pbnm phase approaches the value of five, indicating a transition to a higher symmetry.Note that during the phase transition (between 16.2 and 21 GPa), VA/VB goes beyond five at 16.2 GPa, indicating large pre-transition straining.Detailed reasoning for these phenomena is unclear.While coexistence of two phases may compromise refinement accuracy, simultaneously fitting two phases did not solve this issue in the Pbnm phase.Although the pure high-pressure phase was observed at 21 GPa, the transition likely starts at 16.2 GPa since significant weakening of (111), ( 021), ( 113), (023)/(221), and (133) peaks occurs at this pressure, which is still higher than the transition pressure (12 GPa) predicted by Ross et al. [30,31].The structural deviation of the perovskite (ABX 3 ) structure from its ideal cubic structure is commonly characterized by its BX 6 octahedral titling.The Pbnm perovskite has a simple tilt system of a − a − c + [49,50] (the superscript + and − indicate that successive octahedra along an axis have the same tilt and opposite tilt respectively).Figure 3a shows the GdAlO 3 structure model based on the refinement of the X-ray diffraction at ambient condition.The tilting angles along the a-axis and the b-axis are identical, and successive octahedra along each of these axes have opposite tilt, whereas successive octahedra along the c-axis have the same tilt.The R3c perovskite has a tilt system of a − a − a − [49,50].Figure 3b shows the high-pressure phase structure model of GdAlO 3 based on the refinement of the X-ray diffraction at 21 GPa.There are three identical tilts along the a, b and c-axis, and successive octahedra along each of the axes have opposite tilt.The degree of structure distortion of perovskite can be evaluated using the volumetric ratio (V A /V B ) between AX 12 and BX 6 polyhedron with an ideal value of five for undistorted cubic structure [15].Assuming regular BX 6 octahedra throughout the Pbnm perovskite structure, V A and V B can be calculated through its unit cell dimensions [15,51].Evolution of the V A /V B ratio of the low-pressure phase in GdAlO 3 with pressure is given in Figure 4 shows the V A /V B ratio as a function of pressure (Table 2).As pressure increases, the V A /V B ratio of the Pbnm phase approaches the value of five, indicating a transition to a higher symmetry.Note that during the phase transition (between 16.2 and 21 GPa), V A /V B goes beyond five at 16.2 GPa, indicating large pre-transition straining.Detailed reasoning for these phenomena is unclear.While coexistence of two phases may compromise refinement accuracy, simultaneously fitting two phases did not solve this issue in the Pbnm phase.Although the pure high-pressure phase was observed at 21 GPa, the transition likely starts at 16.2 GPa since significant weakening of (111), ( 021), ( 113), (023)/(221), and ( 133) peaks occurs at this pressure, which is still higher than the transition pressure (12 GPa) predicted by Ross et al. [30,31].[45].** Va and Vb are calculated from the unit cell dimensions.
The compression data (Figure 5) show that the transition from the Pbnm to R3 c phase in GdAlO3 at a pressure between 16.2 GPa and 21 GPa accompanied nearly no volumetric discontinuity.Nevertheless, the space group-subgroup theory indicates that the Pbnm to R3 c transformation is a first-order phase transition [16].Likely, the volumetric discontinuity is so small that it is unrecognizable with the experimental uncertainty of this study.The orthorhombic (Pnma) to rhombohedral (R3 c) transition observed in GdAlO3 and LaGaO3 has a similar unrecognizable volume discontinuity [20,21].* Experimental uncertainty taking the nonhydrostatic pressure due to solidification of pressure medium into account [45].** V a and V b are calculated from the unit cell dimensions.
The compression data (Figure 5) show that the transition from the Pbnm to R3c phase in GdAlO 3 at a pressure between 16.2 GPa and 21 GPa accompanied nearly no volumetric discontinuity.Nevertheless, the space group-subgroup theory indicates that the Pbnm to R3c transformation is a first-order phase transition [16].Likely, the volumetric discontinuity is so small that it is unrecognizable with the experimental uncertainty of this study.The orthorhombic (Pnma) to rhombohedral (R3c) transition observed in GdAlO 3 and LaGaO 3 has a similar unrecognizable volume discontinuity [20,21].from Pbnm to a higher symmetry [30,31].The shaded area shows the pressure range where the Pbnm to R3 c transition may actually happen under non-hydrostatic pressure.The obtained volumetric data of the Pbnm phase were fitted into the third-order Birch-Murnaghan equation of state (EoS) [52]: where P is the pressure applied, V is the volume of the unit cell under pressure, K 0 is the bulk modulus, subscript 0 indicates the ambient pressure, and K 0 is the pressure derivative of bulk modulus.EoSFit 7.0 software [53] was used for fitting the experimental P-V data into the EoS.The fitting was weighted according to the uncertainty of each point [53] and K 0 was fixed at 5.8 based on the previous experimental data by Ross et al. [31].The V 0 and K 0 determined through the fitting are 207.29 ± 0.01 Å 3 and 216 ± 7 GPa, respectively.Due to the very limited data for the high-pressure R3c phase (only two points), no EoS fitting was performed.

Discussion
While temperature influence on the structure of perovskite has been extensively studied, investigations of pressure influence on perovskite structure are still limited.Pressure influence on the Pbnm perovskite structure is twofold.Some structural studies on MgSiO 3 [54,55] and NaMgF 3 [11,56] perovskite showed that the V A /V B ratio of the Pbnm perovskite structure decreases further away from the ideal value of five upon compression, i.e., pressure increases the distortion of the perovskite.Experimental studies of pressure effect on CaGeO 3 perovskite structure presented controversial results.In situ extended X-ray absorption fine structure (EXAFS) study inferred a trend of reduced distortion under high pressure [57], while X-ray diffraction study indicated a reversed pressure effect [58].Controversial results about the pressure influence also exist in theoretical calculations on CaSiO 3 perovskite [59,60].CaSnO 3 is another example of perovskites that show increased distortion with increasing pressure [61].Whether pressure enhances the distortion or reduces it depends on the relative compressibility of the AX 6 octahedra and the BX 12 polyhedron.In the case of GdAlO 3 of this study, the GdO 6 octahedra are less compressible than the AlO 12 polyhedron, yielding an increase in the V A /V B ratio towards five.This is consistent with previous single-crystal X-ray diffraction study on GdAlO 3 by Ross et al. [30,31], although their experiment was limited to the high pressure of 8 GPa and therefore did not observe any phase transition.An ab initio calculation [62] and a single-crystal X-ray diffraction study [63] on another rare-earth aluminate perovskite, YAlO 3 , also show that the AX 6 octahedra are less compressible than the BX 12 polyhedron.All of these align nicely with the model developed by Zhao et al. [17] which shows that in GdFeO 3 -type orthorhombic perovskites with both cations possessing a formal charge of 3+, the structural distortion decreases with increasing pressure and the structure evolves towards a higher symmetry at high pressure.
The bulk modulus (216 GPa) of Pbnm GdAlO 3 derived in this study is higher than those reported previously by Bass et al. [64] using Brillouin spectroscopy (204 GPa) and by Ross et al. [31] using single-crystal X-ray diffraction (191 GPa).In perovskite-structured alkaline earth silicates, germanates, and titanates, a simple trend of a smaller unit formula volume leading to a larger bulk modulus has been observed [65].In the case of rare-earth aluminate perovskite, a summary of the available bulk moduli is listed in Table 3. Figure 6 shows a plot of the bulk moduli at ambient conditions as a function of ionic radius of R 3+ [66] in the rare-earth aluminate perovskite RAlO 3 (R = La, Pr, Nd, Sm, Eu and Gd).The linear trendline defined by the data of LaAlO 3 , PrAlO 3 , NdAlO 3 , SmAlO 3 and EuAlO 3 indicates that the bulk modulus of RAlO 3 increases as the ionic radius of R 3+ decreases.The bulk modulus value of GdAlO 3 derived from the current study agrees well with the general trend of bulk moduli as a function of the ionic radius (Figure 6).More systematic studies on the bulk moduli of the perovskite-structured RAlO 3 aluminates for the entire rare-earth family will be interesting to generalize the influence of ionic radius on bulk modulus of the rare-earth aluminate perovskite.We also realize that the methanol-ethanol pressure medium may yield non-hydrostatic environment when the pressure rises above 10 GPa due to solidification [45].Although enlarged experimental uncertainties because of the non-hydrostatic pressure have been taken into account through weighted EoS fitting, a future study using nearly hydrostatic pressure medium such as helium or neon is more desirable for a better characterization of the compression behavior and the phase transition.  of the non-hydrostatic pressure have been taken into account through weighted EoS fitting, a future study using nearly hydrostatic pressure medium such as helium or neon is more desirable for a better characterization of the compression behavior and the phase transition.[68], MC 2010 [69]), PrAlO3 [70], NdAlO3 [71], SmAlO3 [64] and EuAlO3 [73].
Data for GdAlO3 of this study, JB 1984 [64] and NR 2004 [31] are plotted to show the agreement of these data with the linear trendline.

Conclusions
The previously predicted pressure-induced phase transformation from orthorhombic (Pbnm) to a higher-symmetry structure in GdAlO3 has been observed in in situ X-ray diffraction and the higher-symmetry phase is identified with space group of R3 c with the assistant of DFT calculations.(JZ 2004 [68], MC 2010 [69]), PrAlO 3 [70], NdAlO 3 [71], SmAlO 3 [64] and EuAlO 3 [73].Data for GdAlO 3 of this study, JB 1984 [64] and NR 2004 [31] are plotted to show the agreement of these data with the linear trendline.

Conclusions
The previously predicted pressure-induced phase transformation from orthorhombic (Pbnm) to a higher-symmetry structure in GdAlO 3 has been observed in in situ X-ray diffraction and the higher-symmetry phase is identified with space group of R3c with the assistant of DFT calculations.The X-ray diffraction patterns indicate that the structural phase transformation occurs between 16 and 21 GPa, although the volumetric variations V A /V B of the Pbnm phase suggest the transition to a higher symmetry may apparently start at around 14 GPa.The structural phase transformation is a first-order transition according to the space groups but the volumetric discontinuity is indistinguishable with the experimental uncertainty of the current study.A bulk modulus (K 0 ) of 216 ± 7 GPa with a fixed derivative bulk modulus (K 0 ) at 5.8 was obtained for the Pbnm phase according to the third-order Birch-Murnaghan equation of state.The bulk modulus derived in this study is higher than the previously reported values but is in good agreement with the linear trend of increase in bulk modulus as the ionic radius of R 3+ in rare-earth aluminate perovskite RAlO 3 decreases.

Figure 1 .
Figure 1.In situ synchrotron X-ray diffraction patterns of GdAlO3 at different pressures as indicated and room temperature (λ = 0.3757 Å).Red circles indicate the characteristic peaks that disappear when Pbnm to R3 c phase transition occurs.

Figure 1 .
Figure 1.In situ synchrotron X-ray diffraction patterns of GdAlO 3 at different pressures as indicated and room temperature (λ = 0.3757 Å).Red circles indicate the characteristic peaks that disappear when Pbnm to R3c phase transition occurs.

Crystals 2023 , 13 Figure 2 .
Figure 2. Result of Rietveld refinement of the diffraction pattern of GdAlO3 at 21 GPa with (a) Pbnm, (b) I4/mcm and (c) R3 c space groups.Crosses are experimentally observed data, red solid line is calculated data, vertical bars are locations of expected diffraction peaks, and blue line at the bottom is the difference between the observed and calculated data.

Figure 2 .
Figure 2. Result of Rietveld refinement of the diffraction pattern of GdAlO 3 at 21 GPa with (a) Pbnm, (b) I4/mcm and (c) R3c space groups.Crosses are experimentally observed data, red solid line is calculated data, vertical bars are locations of expected diffraction peaks, and blue line at the bottom is the difference between the observed and calculated data.

Figure 3 .Table 2 .
Figure 3. Crystal structure of GdAlO3 at (a) ambient pressure (Pbnm space group, a − a − c + tilt system), and (b) 21 GPa (R3 c space group, a − a − a − tilt system).Solid black lines indicate unit cell.Small red dots represent oxygen atoms and large pink spheres gadolinium atoms.Grey areas denote AlO6 octahedra.Table 2. Unit cell parameters of GdAlO3 measured in the diamond anvil cell from 1 bar to 23 GPa.The figures in parentheses represent 1 esd of the last decimal place shown.P (GPa) σ * (GPa) a (Å) b (Å) c (Å) V (Å 3 ) Va/Vb **

Figure 3 .
Figure 3. Crystal structure of GdAlO 3 at (a) ambient pressure (Pbnm space group, a − a − c + tilt system), and (b) 21 GPa (R3c space group, a − a − a − tilt system).Solid black lines indicate unit cell.Small red dots represent oxygen atoms and large pink spheres gadolinium atoms.Grey areas denote AlO 6 octahedra.Crystals 2023, 13, x FOR PEER REVIEW 7 of 13

Figure 4 .
Figure 4. Variations of VA/VB of the Pbnm (solid symbols) and R3 c (open symbols) phase of GdAlO3 as a function of pressure.The broken line indicates the ideal value of VA/VB for undistorted cubic perovskite structure.The dotted line indicates the pressure of the predicted structural transition tofrom Pbnm to a higher symmetry[30,31].The shaded area shows the pressure range where the Pbnm to R3 c transition may actually happen under non-hydrostatic pressure.

Figure 4 .
Figure 4. Variations of V A /V B of the Pbnm (solid symbols) and R3c (open symbols) phase of GdAlO 3 as a function of pressure.The broken line indicates the ideal value of V A /V B for undistorted cubic perovskite structure.The dotted line indicates the pressure of the predicted structural transition to from Pbnm to a higher symmetry [30,31].The shaded area shows the pressure range where the Pbnm to R3c transition may actually happen under non-hydrostatic pressure.

Figure 4 .
Figure 4. Variations of VA/VB of the Pbnm (solid symbols) and R3 c (open symbols) phase of GdAlO3 as a function of pressure.The broken line indicates the ideal value of VA/VB for undistorted cubic perovskite structure.The dotted line indicates the pressure of the predicted structural transition to

Figure 5 .
Figure 5. Variation of the volume per formula unit of GdAlO3 perovskite (triangles: the Pbnm phase, squares: the R3 c phase) as a function of pressure.The experimental errors are either indicated by the bars or less than the size of the symbols.The solid line represents the third-order Birch-Murnaghan EoS fitting for the data of the Pbnm phase (triangles).The obtained volumetric data of the Pbnm phase were fitted into the third-order Birch-Murnaghan equation of state (EoS)[52]:

Figure 5 .
Figure 5. Variation of the volume per formula unit of GdAlO 3 perovskite (triangles: the Pbnm phase, squares: the R3c phase) as a function of pressure.The experimental errors are either indicated by the bars or less than the size of the symbols.The solid line represents the third-order Birch-Murnaghan EoS fitting for the data of the Pbnm phase (triangles).

Table 1 .
Results of Rietveld refinement of the diffraction pattern of GdAlO 3 at 21 GPa using Pbnm, I4/mcm and R3c space groups.

Table 2 .
Unit cell parameters of GdAlO 3 measured in the diamond anvil cell from 1 bar to 23 GPa.The figures in parentheses represent 1 esd of the last decimal place shown.

Table 3 .
Summary of the available bulk moduli (K) of rare-earth aluminate perovskite RAlO 3 .

Table 3 .
Summary of the available bulk moduli (K) of rare-earth aluminate perovskite RAlO3.