Pressure-Driven Phase Transition in InPO₄: The Elastic Response of CrVO₄-Type, Scheelite, and Zircon Polymorphs

Abstract

In this work, we present a theoretical study of InPO₄ under pressure. Total-energy calculations based on density functional theory were performed to explore the crystal structure of InPO₄ in light of the recent X-ray diffraction characterization of this compound under pressure. A phase coexistence was observed above 10 GPa, involving the ambient-pressure CrVO₄-type structure and the high-pressure scheelite and zircon phases. Therefore, the previously performed analysis of InPO₄ behavior under pressure is extended by simulating X-ray spectra and interplanar distances for the three polymorphs. In addition, the elastic behavior of the three phases is analyzed to assess the elastic stability of InPO₄ under pressure and to compute the mechanical properties and elastic anisotropy. Our findings significantly extend previous experimental results on the compressibility of InPO₄, which were limited to the ambient-pressure phase. Moreover, our results unambiguously reveal a marked difference in the elastic properties of the scheelite and zircon phases under pressure, showing that the zircon phase is elastically unstable at high pressures. This suggests that the reported coexistence of phases may result from kinetic barriers or from non-hydrostatic conditions within the diamond anvil cell caused by the pressure-transmitting medium.

1. Introduction

One of the main characteristics of the ABO₄ compounds is their remarkable structural versatility, as they can crystallize in a wide range of crystal structures. Their polymorphism is determined by several physical and chemical parameters, among which the ionic radii of the cations and the anion ($r_A$, $r_B$, $r_{\mathrm{O}}$), the atomic nature of the elements that constitute the system, and the coordination number of the cations with oxygen are particularly important [1]. Thus, many of these compounds crystallize in specific structures, such as the zircon [2], scheelite [3], fergusonite [4], wolframite [5], and CrVO₄-type [6] structures, among others. The common thread among these structures is captured by Bastide’s diagram [1], in which the structures of the compounds ABO₄ are arranged according to the aspect ratios of their ionic radii [(x, y): ($r_A$/$r_{\mathrm{O}}$, $r_B$/$r_{\mathrm{O}}$)]. In addition, the north-east rule in Bastide’s diagram helps predict the pressure-driven phase transition in ABO₄ compounds [7].

Among the structures mentioned, the orthorhombic CrVO₄-type structure [Space Group (SG): $Cmcm$, No. 63, Z = 4] is a characteristic polymorph of vanadates AVO₄ [8], phosphates APO₄ [9], chromates ACrO₄, sulfates ASO₄, and selenates ASeO₄ [10]. Within this structure, there are four non-equivalent Wyckoff positions (WP): cation A occupies 4a (0, 0, 0), cation B occupies 4c (0, y, 1/4), and oxygen atoms are distributed among 8f (0, y, z) and 8g (x, y, 1/4). The crystal structure consists of chains of AO₆ octahedra nearly aligned with the z axis; these distorted octahedra have two non-equivalent bond distances ($d_{A-\mathrm{O}1}$ in the equator and $d_{A-\mathrm{O}2}$ at the apex). These chains coexist with BO₄-distorted tetrahedra, which also exhibit two non-equivalent bond distances ($d_{B-\mathrm{O}1}$, $d_{B-\mathrm{O}2}$). According to the literature, the phosphates that crystallize in the CrVO₄-type structure at ambient pressure are TiPO₄, VPO₄, CrPO₄, InPO₄, and TlPO₄ [10]. The CrVO₄-type crystal structure of InPO₄ is shown in Figure 1a, which details the connections between polyhedra and the internal interatomic distances.

Indium phosphate was recently studied under pressure; experiments were conducted using synchrotron-based angle-dispersive X-ray diffraction and Raman scattering in a Mao-Bell-type diamond anvil cell (DAC) up to 31.5 GPa [12]. It was reported that the CrVO₄-type structure remains stable up to 10 GPa; beyond 12.3 GPa, the Raman spectra and X-ray diffraction patterns showed signatures of a structural phase transition, with CrVO₄-type, scheelite (SG: I4₁/a, No. 88, Z = 4), and zircon (SG: I4₁/amd, No. 141, Z = 4) phases coexisting up to the highest pressure reached in the DAC. Phase coexistence has been previously observed in ABO₄ compounds, such as FeVO₄ [13]. However, subsequent studies did not observe this phase coexistence [14].

For InPO₄, the observed tetragonal phases consist of InO₈ dodecahedra and PO₄ tetrahedra. In zircon, the WPs for In, P, and O are 4b (0, 0.25, 0.375), 4a (0, 0.75, 0.125), and 16h (0, y, z), respectively. Figure 1b shows that the structure is formed by alternating edge-sharing InO₈ dodecahedra and PO₄ tetrahedra, which form chains parallel to the c axis. In the scheelite structure, the WPs for In, P, and O are 4b (0, 0.25, 0.625), 4a (0, 0.25, 0.125), and 16f (x, y, z), respectively. According to Figure 1b, the InO₈ units share edges and corners with other InO₈ units and with PO₄ tetrahedra. According to Figure 1b,c, there is only one P–O interatomic distance ($d_{\mathrm{P}-\mathrm{O}}$) and two In–O interatomic bond distances ($d_{\mathrm{In}-\mathrm{O}}$) in the tetragonal structures.

According to the north-east rule of the Bastide diagram, the CrVO₄-type structure could undergo a pressure-driven phase transition to the zircon phase and subsequently to a scheelite phase [1,10]. A previous first-principles study investigated the structural and vibrational changes of InPO₄ and TiPO₄ under pressure and reported such a phase transition [9]. Another theoretical report observed the same behavior for TiSiO₄ [15]. Therefore, the experimental results for InPO₄ [12] are of utmost importance, since many ABO₄ compounds with a CrVO₄-type structure undergo pressure-induced phase transitions to a monoclinic structure [6,16,17,18]. Furthermore, the new results offer an opportunity to examine the mechanical properties of this compound more thoroughly, beyond the bulk modulus.

In this work, we explore the structural changes in InPO₄ using first-principles calculations up to 20 GPa, a pressure range that corresponds to nearly hydrostatic behavior [19,20,21] for the pressure-transmitting medium (4:1 methanol-ethanol mixture) used in the experiments within the DAC performed in Ref. [12]. Given the reported experimental results, we analyze the CrVO₄-type structure up to 10 GPa and the high-pressure phases between 10 and 22 GPa, comparing our results with available experimental data. In addition, we analyze the elastic and mechanical properties, including anisotropic elasticity, to gain a deeper understanding of InPO₄ at high pressures. The evolution of the mechanical properties is examined by taking into account the experimental results obtained from the equation of state (EOS) of the low-pressure phase with CrVO₄-type structure [12].

The paper is organized as follows: the computational details are presented in the next section. The results for the crystal structure, elastic constants, mechanical properties, and elastic anisotropy are reported in Section 3.1, Section 3.2, Section 3.3, and Section 3.4, respectively. Finally, a summary and the conclusions are provided in Section 4.

2. Computational Details

First-principles calculations were performed within the framework of the density functional theory (DFT) [22] and the projector-augmented wave (PAW) [23,24] method as implemented in the Vienna Ab initio Simulation Package (VASP) [25]. We have considered 13 valence electrons for In (5$s^2$4$d^{10}$5$p^1$), five for P (3$s^2$3$p^3$), and six for O (2$s^2$2$p^4$) atoms in the PAW pseudo-potential. A plane-wave energy cutoff of 520 eV was used to ensure a high precision in our calculations. The exchange-correlation energy has been described within the generalized-gradient approximation (GGA) in the Perdew–Burke–Ernzerhof for solids (PBEsol) formulation [26].

The self-consistent calculations were performed with the Monkhorst–Pack scheme for the Brillouin-zone (BZ) integrations [27] with meshes 4 × 3 × 3, 4 × 4 × 4, and 4 × 4 × 2, corresponding to sets of 36, 40, and 32 special k-points in the irreducible BZ for the CrVO₄-type, zircon, and scheelite phases, respectively. In the relaxed equilibrium configuration, the forces are less than one meV/Å per atom in each Cartesian direction. The calculations of the elastic constants have been performed with k-point meshes 8 × 6 × 6, 8 × 8 × 8, and 8 × 8 × 4, a plane-wave energy cutoff of 570 eV, and a POTIM parameter of 0.010. Highly converged results for forces are required to obtain the elastic constants under pressure [28]. The elastic tensor has been determined by performing six finite distortions of the lattice and deriving the elastic constants from the strain–stress relationship [29]. The processing of the elastic constant data calculated with VASP was performed using the VASPKIT program [30]. Our computed X-ray diffraction patterns were generated with the VESTA program [11] using the experimental $\lambda$ wavelength ($\lambda$ = 0.6125 Å). The simulated X-ray peak intensities are displayed on a relative scale from 0 to 100, where the most intense peak of each phase is set to 100 arbitrary units. The relative peak intensities are proportional to the square of the structure factor: $I\propto |F_{hkl}{|}^2$.

3. Results and Discussion

3.1. Crystal Structure

The theoretically calculated X-ray diffraction (XRD) patterns of InPO₄ in the orthorhombic CrVO₄-type structure at ambient pressure are shown in Figure 2a. The simulated diffraction pattern reproduces the relative peak positions and intensities reported experimentally for this phase [12]. Our computed lattice parameters [9] are a = 5.3355 Å, b = 8.0454 Å, and c = 6.8255 Å (V = 293 ų), which are in good agreement with the experimental values a = 5.3212(10) Å, b = 8.0096(8) Å, and c = 6.7955(12) Å (V = 289.62 ų) [12]. The differences between the theoretical and experimental results are less than 1% (≈1%), confirming the reliability of the structural description of InPO₄ used in this work.

As shown in Figure 2a, the most intense reflections occur at approximately 7.89, 8.73, 13.11, and 14.68 $2\theta$ degrees, which can be indexed to the (110), (020), (112), and (130) crystallographic lattice planes, respectively, characteristic of the orthorhombic CrVO₄-type structure of InPO₄ at ambient pressure. These results are in excellent agreement with the Rietveld-refined X-ray diffraction pattern reported by Dwivedi et al. for InPO₄ obtained using synchrotron radiation with a wavelength of $\lambda$ = 0.6125 Å [12].

As pressure increases from atmospheric pressure to ≈20 GPa, a systematic evolution of the diffraction peaks is observed. These changes are characterized primarily by a continuous shift of the reflections toward higher $2\theta$ angles, indicating a progressive reduction in the interplanar spacings $d_{hkl}$. This behavior is consistent with the reduction of the lattice parameters induced by external pressure [9,12].

Throughout the entire pressure range investigated, the characteristic diffraction peaks of the CrVO₄-type structure persist, with no new reflections appearing or existing ones abruptly disappearing. This suggests that the orthorhombic structure of InPO₄ remains stable over the pressure range studied. However, slight variations in the relative intensities of some peaks are observed, which may be associated with anisotropic changes in the lattice parameters and internal atomic positions under compression. The change in internal parameters with pressure is shown in Figure 3, indicating that the most significant changes in the crystal structure are associated with the In-O polyhedra across the three polymorphs. In particular, despite slight changes in the interatomic distances of the PO₄ tetrahedron ($d_{\mathrm{P}-\mathrm{O}n}$), its polyhedral volume ($V_{{\mathrm{PO}}_4}$) remains almost constant, due to changes in its internal angles (${\theta }_n$ and ${\varphi }_n$), which are reflected in the distortion parameter (${\Delta }_d$).

The evolution of the interplanar distances $d_{hkl}$ with pressure for the most intense peaks is shown in Figure 2c. The most pronounced changes in the experimental results for InPO₄ occur for the (110) and (112) lattice planes [12]. A shift in these interplanar distances was already observed in the orthorhombic phase III of CrVO₄ under pressure, but over a shorter pressure range [6]. In the CrVO₄-type structure of InVO₄, the most relevant changes under pressure were observed for the (020), (112), and (130) lattice planes [16].

Given the phase coexistence above 12 GPa observed in the experiments [12], Figure 2b shows the simulated X-ray diffraction patterns of the zircon and scheelite phases up to 20.24 GPa. Figure 2c shows the interplanar distance $d_{hkl}$ of the most representative planes of both phases as a function of pressure. As clearly shown, the change in $d_{hkl}$ is most significant for the CrVO₄-type structure, which is due to the fact that the elastic constants and mechanical properties of this phase are smaller than those of zircon and scheelite.

In the experimental results from Ref. [12], the $hkl$ indices for the Bragg reflections are not reported, making it difficult to analyze our data in relation to those results. Furthermore, to our knowledge, there are no experimental X-ray diffraction studies of other phosphates that exhibit a pressure-driven phase transition from CrVO₄-type to zircon or scheelite and could be used to correlate with our results. However, to support our results, we compare the scheelite X-ray diffraction pattern of InPO₄ with that of SrWO₄ [31], which is one of the most studied scheelite wolframates [1]. To compare our simulated X-ray diffraction pattern, we used the wavelength $\lambda$ = 1.5406 Å employed in Ref. [31] and found good agreement with the relative positions of the diffraction peaks of scheelite SrWO₄. The most intense reflections occur at 8.31, 12.63, and 20.16 $2\theta$ degrees for the (101), (112), and (204) crystallographic lattice planes at 12 GPa, respectively. As in scheelite compounds, the most significant changes occur along the [001] direction because the crystal is better packed in the $xy$ plane.

The same procedure was successfully applied to the zircon phase by comparing our simulated X-ray diffraction pattern with experimental X-ray characterization of zircon NdVO₄ at a wavelength $\lambda$ = 0.6199 Å [32]. In this case, the most intense reflections occur at 8.07, 10.81, 14.24, and 20.96 $2\theta$ degrees for the (101), (200), (112), and (312) crystallographic lattice planes at 12.33 GPa, respectively. In this case, the structural changes are more pronounced in the lattice parameter a, as will be seen in the following section.

3.2. Elastic Constants

The elastic behavior of the orthorhombic CrVO₄-type polymorph of InPO₄ is described by nine independent elastic constants ($c_{11}$, $c_{22}$, $c_{33}$, $c_{44}$, $c_{55}$, $c_{66}$, $c_{12}$, $c_{13}$, $c_{23}$) [33]. $c_{11}$, $c_{22}$, and $c_{33}$ are associated with longitudinal compression, $c_{12}$, $c_{13}$, and $c_{23}$ with transverse expansion, and $c_{44}$, $c_{55}$, and $c_{66}$ with pure shear [34]). The computed elastic constants and their pressure coefficients for InPO₄ in the CrVO₄-type structure (at atmospheric pressure), zircon (at 12.3 GPa), and scheelite (at 13 GPa) phases are presented in

Table 1. Elastic constants, $c_{ij}$ (in GPa), of InPO₄ for the CrVO₄-type structure, zircon, and scheelite at the specified pressure, P (in GPa). Pressure coefficients, $d{c}_{ij}/dP$, for each $c_{ij}$ appear in brackets.

	${\mathit{c}}_{11}$	${\mathit{c}}_{12}$	${\mathit{c}}_{13}$	${\mathit{c}}_{16}$	${\mathit{c}}_{22}$	${\mathit{c}}_{23}$	${\mathit{c}}_{33}$	${\mathit{c}}_{44}$	${\mathit{c}}_{55}$	${\mathit{c}}_{66}$	P
CrVO4-type	245.74	58.92	80.43		107.54	40.43	151.64	55.98	67.61	41.82	0.01
	304.12	65.53	124.75		102.62	69.44	200.34	64.24	99.10	39.07	10.15
($d{c}_{ij}/dP$)	(5.73)	(0.60)	(4.35)		(−0.55)	(2.84)	(4.79)	(0.78)	(3.08)	(−0.27)
Zircon	365.21	68.99	157.91				451.16	91.20		16.89	12.33
($d{c}_{ij}/dP$)	(8.09)	(2.35)	(2.33)				(4.27)	(0.89)		(−0.59)
Scheelite	386.95	179.39	168.20	−15.49			259.86	72.45		116.22	13.03
($d{c}_{ij}/dP$)	(4.77)	(2.47)	(2.99)	(0.81)			(6.74)	(1.92)		(1.74)

. Taken together, these $c_{ij}$ describe the stability and anisotropic response of the crystal to external stresses and indicate which planes and directions are more prone to deformation. Therefore, we assess the stability of the system using the criteria for orthorhombic systems, which require the following: $c_{11}>0$, $c_{11}{c}_{22}>c_{12}^2$, $c_{11}{c}_{22}{c}_{33}+2c_{12}{c}_{13}{c}_{23}-c_{11}{c}_{23}^2-c_{22}{c}_{13}^2-c_{33}{c}_{12}^2>0$, $c_{44}>0$, $c_{55}>0$, and $c_{66}>0$ [35]. Our results in Table 1 for the CrVO₄-type structure at atmospheric pressure confirm that these conditions are satisfied.

At ambient pressure, the relation $c_{11}>c_{33}>c_{22}$ is observed, indicating that the x axis is the most resistant to longitudinal compression. This trend was also observed in the CrVO₄-type structure of FeVO₄ [18]. This behavior aligns with trends in the structural parameters and reflects differences in the crystal lattice’s packing efficiency [9,12]. The transverse expansion constants indicate that $c_{13}>c_{12}>c_{23}$, meaning the z axis is more prone to deformation under [010] stress. Regarding the shear elastic constants, $c_{55}>c_{44}>c_{66}$, indicating that the $xy$ plane is the most susceptible to deformation.

Under hydrostatic pressure, the extended stability criteria for the orthorhombic CrVO₄-type structure are as follows [36]: ${\tilde{c}}_{11}>0$, ${\tilde{c}}_{11}{\tilde{c}}_{22}>{\tilde{c}}_{12}^2$, ${\tilde{c}}_{11}{\tilde{c}}_{22}{\tilde{c}}_{33}+2{\tilde{c}}_{12}{\tilde{c}}_{13}{\tilde{c}}_{23}-{\tilde{c}}_{11}{\tilde{c}}_{23}^2-{\tilde{c}}_{22}{\tilde{c}}_{13}^2-{\tilde{c}}_{33}{\tilde{c}}_{12}^2>0$, ${\tilde{c}}_{44}>0$, ${\tilde{c}}_{55}>0$, and ${\tilde{c}}_{66}>0$. Here, ${\tilde{c}}_{ii}=c_{ii}-P$ (i = 1, …, 6), ${\tilde{c}}_{12}=c_{12}+P$, ${\tilde{c}}_{13}=c_{13}+P$, and ${\tilde{c}}_{23}=c_{23}+P$. According to the $c_{ij}$ values in Table 1 at 10.15 GPa, the CrVO₄-type structure satisfies the stability criterion, indicating elastic stability under pressure. Previous results on the phonon spectrum of InPO₄ also show that the CrVO₄-type structure is dynamically stable from atmospheric pressure [9] up to 10 GPa (not published).

According to Table 1 and Figure 4, the longitudinal compression elastic constants, $c_{11}$ and $c_{33}$, show the largest increases with pressure, reflecting improved packing efficiency. In contrast, the decrease in $c_{22}$ aligns with the high sensitivity of the lattice parameter b to the notable reduction in the equatorial distance $d_{\mathrm{In}-\mathrm{O}1}$ relative to the apical distance in the InO₆ polyhedra, see Figure 1a and Figure 3a. For the transverse compression constants, although all exhibit positive pressure coefficients, $c_{13}$ and $c_{23}$ show significantly larger pressure coefficients than $c_{12}$, reinforcing the idea that the y axis controls a large part of the mechanical anisotropy of the CrVO₄-type structure under pressure. On the other hand, although the PO₄ tetrahedra do not undergo such pronounced deformations, they display significant angular variations that enable the rotation of the octahedra, indirectly contributing to the strong anisotropy observed along the y axis [9], which ultimately leads to the pressure-induced phase transition [12].

As shown in Figure 4, the elastic constants of the tetragonal zircon and scheelite high-pressure phases were computed between 10 and 22 GPa. The tetragonal high-pressure phases belong to different Laue classes: $4/m$ for scheelite and $4/mmm$ for zircon. Both structures have the same six elastic constants ($c_{11}$, $c_{33}$, $c_{44}$, $c_{66}$, $c_{12}$, and $c_{13}$), but scheelite, due to its Laue class, has a seventh elastic constant, $c_{16}$, which indicates coupling between strain and shear stresses. The significance of the first six elastic constants is similar to that of the CrVO₄-type structure, with the difference that in these phases $c_{11}$ = $c_{22}$, $c_{44}$ = $c_{55}$, and $c_{13}$ = $c_{23}$ [34]. Table 1 presents the calculated values for the zircon (12.33 GPa) and scheelite (13.03 GPa) phases, together with their pressure coefficients.

The elastic stability criteria for zircon under pressure require that [36]: ${\tilde{c}}_{44}>0$, ${\tilde{c}}_{66}>0$, ${\tilde{c}}_{11}-\left|{\tilde{c}}_{12}\right|>0$, and ${\tilde{c}}_{33}({\tilde{c}}_{11}+{\tilde{c}}_{12})-2{\tilde{c}}_{13}^2>0$. Based on the $c_{ij}$ and $d{c}_{ij}/dP$ values in Table 1, the zircon phase is elastically stable up to 15.2 GPa; beyond this pressure, the stability condition ${\tilde{c}}_{66}>0$ is no longer satisfied. These results may be related to the small slope of one acoustic phonon branch along the M-$\mathsf{\Gamma }$ path in zircon’s phonon spectrum at low pressures. Under conditions of non-coexistence of phases, the zircon phase is stable only at low pressures and within a relatively small pressure range [9].

It is observed that $c_{33}>c_{11}$ in the zircon phase, indicating that the z axis is the most resistant to longitudinal compression. Nevertheless, when examining the pressure coefficients, the [100] direction emerges as the most prominent. For the $c_{ij}$ elastic constants, $c_{13}$ is much larger than $c_{12}$. However, both show similar pressure coefficients, indicating that the pressure’s effect on the y and z axes is nearly the same for transverse expansion. The shear elastic constant $c_{44}>c_{66}$ across the entire pressure range considered.

For scheelite under pressure, the stability criteria require [36]: ${\tilde{c}}_{44}>0$, ${\tilde{c}}_{11}-\left|{\tilde{c}}_{12}\right|>0$, ${\tilde{c}}_{33}({\tilde{c}}_{11}+{\tilde{c}}_{12})-2{\tilde{c}}_{13}^2>0$, and ${\tilde{c}}_{66}({\tilde{c}}_{11}-{\tilde{c}}_{12})-2c_{16}^2>0$. Based on the results in Table 1, these conditions are satisfied across the pressure range considered. We also observe that $c_{11}>c_{33}$, indicating that the x axis is the most resistant to longitudinal compression, because the scheelite phase is more compact in the $xy$ plane than along the [001] direction [28]. This behavior is also reflected in the higher value of $c_{12}$ compared to $c_{13}$, indicating that the scheelite phase is more resistant to deformation along the [010] direction than along the [001] direction when stress is applied along the x axis. The fact that $c_{66}>c_{44}$ reflects the material’s strength in the $xy$ plane.

3.3. Mechanical Properties

The bulk (B) and shear (G) modulus were computed using the Hill average [28], which is the arithmetic mean of the Voigt (V) [37] and Reuss (R) [38] bounds. The bulk modulus in the Voigt and Reuss methods is computed as follows: $9B_{\mathrm{V}}=(c_{11}+c_{22}+c_{33})+2(c_{12}+c_{13}+c_{23})$, and $B_{\mathrm{R}}^{-1}=(s_{11}+s_{22}+s_{33})+2(s_{12}+2s_{13}+2s_{23})$, respectively. The shear modulus is obtained with the following equations: 15$G_{\mathrm{V}}=(c_{11}+c_{22}+c_{33})-(c_{12}+c_{13}+c_{23})+3(c_{44}+c_{55}+c_{66})$, and $15G_{\mathrm{R}}^{-1}$ = 4($s_{11}$ +$s_{22}$+ $s_{33}$) − 4($s_{12}$ + $s_{13}$ + $s_{23}$) + 3($s_{44}$ + $s_{55}$ + $s_{66}$), respectively, while $B=(B_{\mathrm{V}}+B_{\mathrm{R}})/2$ and $G=(G_{\mathrm{V}}+G_{\mathrm{R}})/2$ in the Hill average. The Young’s modulus, E, is computed with $E=9BG/(3B+G)$, and Poisson’s ratio, $\nu$, with $\nu =(3B-2G)/(6B+2G)$. The Vickers hardness is obtained with the Tian [39] approximation: $H_{\mathrm{V}}$ = 0.92 $k^{1.137}$$G^{0.708}$, where k is the inverse of the Pugh’s ratio ($B/G$), $k=G/B$.

In the experimental results, the bulk modulus was determined by fitting the measured $P-V$ data to the third-order Birch-Murnaghan EOS [12]. The reported value was B = 97 ± 6 GPa, with its pressure derivative $B^{\prime }$ = 7 ± 3. This is the largest value reported for phosphates with a CrVO₄-type structure, namely TiPO₄ (72.93 GPa), VPO₄ (79.16 GPa), and TlPO₄ (53.57 GPa) [9]. Note that B increases with the atomic number of the A cation within a period and decreases from one period to another for the same family in the periodic table. Such behavior has been observed in other ABO₄ compounds, including scheelites [28,40] and wolframites, with some variations [5,41,42]. The result obtained from the Hill procedure at ambient pressure is B = 88.56 GPa, close to the lower limit of the experimental values. According to Figure 5a, the bulk modulus increases with pressure, indicating a progressive improvement in the crystal structure’s packing efficiency. This hardening is particularly pronounced in the high-pressure phases (almost twice the value of the low-pressure phase at 10 GPa), with a slightly higher value in scheelite than in zircon, as expected given their locations on the Bastide diagram [1].

Figure 5b shows that the CrVO₄-type phase of InPO₄ has a shear modulus of G = 53 GPa at atmospheric pressure, which is small compared with diamond (G = 533(3) GPa [43]). However, it is larger than that of more than 20 scheelite ABO₄ compounds [44]. In comparison with other compounds with a CrVO₄-type structure, it is similar to FeVO₄ (G = 52.58 GPa) [28] but larger than TlVO₄ (G = 25.13 GPa) [17]. It was observed that G increases with pressure, reaching a maximum of 58.13 GPa at 10.15 GPa, a 10% increase, which is half that of scheelite in the same pressure range. By contrast, the behavior of zircon is completely different because G depends on $c_{66}$, which has a negative pressure coefficient. Therefore, the softening of G above 16.1 GPa reveals that the zircon structure does not satisfy the extended stability criteria above this pressure, since ${\tilde{c}}_{66}<0$. These findings reinforce the notion that the zircon phase is not stable at such high pressures. The remaining mechanical properties (E, $\nu$, $B/G$, and $H_{\mathrm{V}}$) depend on the relation between B and G, so the softening of G above 16.1 GPa will affect these properties. In particular, E follows a similar trend to G.

The Poisson’s ratio ($\nu$) of ceramics and semiconductors typically ranges from 0.25 to 0.35 under atmospheric conditions [45]. Covalent materials usually have $\nu$ around 0.1, while ionic ones typically have $\nu$ = 0.25 [46]. The CrVO₄-type structure of InPO₄ has $\nu$ = 0.251 at atmospheric pressure, which lies at the lower limit associated with its ionic character. Since $\nu$ measures the resistance of a material to volume change (B) relative to the resistance to shape change (G) [45], it is closely related to Pugh’s ratio ($B/G$). A high $B/G$ is associated with ductility, whereas a low $B/G$ is associated with brittleness, with $B/G$ = 1.75 as the critical value separating brittle and ductile materials [47]. Our results indicate that the CrVO₄-type structure has a value of 1.673 at ambient pressure, which increases with pressure and even reaches more than twice this value in the scheelite phase at high pressures. The trends in Figure 5e indicate that the ductility of InPO₄ increases with pressure for the CrVO₄-type and scheelite phases, changing the brittle nature of InPO₄ at atmospheric pressure. In contrast, the zircon phase exhibits the opposite behavior, decreasing its ductility with pressure.

The Vickers hardness ($H_{\mathrm{V}}$) is inversely related to Pugh’s ratio in Tian’s model [39]. As a result, $H_{\mathrm{V}}$ varies inversely with $B/G$; see Figure 5f. ABO₄ compounds are not characterized by high Vickers hardness; a clear example is the scheelite compounds, which have $H_{\mathrm{V}}$ values below 11 GPa at atmospheric pressure [44]. By comparison, many other compounds have higher $H_{\mathrm{V}}$ [48], with diamond having $H_{\mathrm{V}}$ = 96 GPa [49]. Despite this, InPO₄ has a larger $H_{\mathrm{V}}$ value than many other ABO₄ compounds at atmospheric pressure [44], see Figure 5f. Even at 10 GPa, the Vickers hardness of the scheelite phase of InPO₄ is higher than that of 25 other scheelites at ambient pressure [44], whose hardness will decrease at 10 GPa. The Vickers hardness results are consistent with those observed for the other mechanical properties: lower $H_{\mathrm{V}}$ values correspond to greater resistance to volume change relative to resistance to shape change. Diamond is the best example of the opposite behavior, with a lower B value [439.2(9) GPa] than G [533(3) GPa] [43].

3.4. Elastic Anisotropy

Deformations in the crystal lattice can lead to imperfections like vacancies, interstitial sites, and dislocations. When deformation reaches the microfracture stage, twinning, stacking faults, or other imperfections may form. These all lead to a broadening of the diffraction lines. Therefore, elastic anisotropy has significant implications for engineering science and crystal physics because it often causes microcrack formation in materials [50]. Note that this broadening does not appear in the simulated X-ray spectra of Figure 2. To address this behavior, DFT calculations could be performed using a supercell approach, with a crystal structure n times the size of the conventional cell to account for structural defects [51]. An alternative method would involve artificial intelligence for a specific application, such as data-driven and machine-learning approaches [52]. However, both options are beyond the scope of this study.

In this work, we analyzed several anisotropic factors that quantify the degree of anisotropy in atomic bonding across different crystallographic planes. For orthorhombic systems, there are three shear anisotropic factors: $A_1$ for the (100) shear plane between the [011] and [010] directions, $A_1=4c_{44}/(c_{11}+c_{33}-2c_{13})$; $A_2$ for the (010) shear plane between the [101] and [001] directions, $A_2=4c_{55}/(c_{22}+c_{33}-2c_{23}$; and $A_3$ for the (001) shear plane between the [110] and [010] directions, $A_3=4c_{66}/(c_{11}+c_{22}-2c_{12})$. The shear anisotropic factors $A_1$, $A_2$, and $A_3$ must equal one for isotropic crystals; any deviation from unity measures the degree of elastic anisotropy. According to Figure 6a, the most significant change under pressure occurs in the $A_2$ shear anisotropic factor for the CrVO₄-type structure and in $A_3$ for zircon, with $A_2$ showing the largest pressure dependence. The change in $A_2$ is related to the low resistance to compression of the CrVO₄-type structure in the [010] direction. In the scheelite phase, $A_1$ and $A_3$ are close to the ideal value of 1. Such behavior indicates that the anisotropy related to the shear anisotropic factors would decrease significantly after a phase transition to CrVO₄-type → scheelite, which would stabilize the InPO₄ at high pressures.

Other important anisotropic factors include the universal anisotropy index, $A^U$ [53], and the percentage of elastic anisotropy in B, $A_B$, and in G, $A_G$ [54]. These indices can be calculated as follows: $A^U=5{\displaystyle \frac{G_{\mathrm{V}}}{G_{\mathrm{R}}}}+{\displaystyle \frac{B_{\mathrm{V}}}{B_{\mathrm{R}}}}-6$, $A_B={\displaystyle \frac{B_{\mathrm{V}}-B_{\mathrm{R}}}{B_{\mathrm{V}}+B_{\mathrm{R}}}}$, and $A_G={\displaystyle \frac{G_{\mathrm{V}}-G_{\mathrm{R}}}{G_{\mathrm{V}}+G_{\mathrm{R}}}}$. For isotropic crystals, $A_B=A_G=A^U=0$; any deviation indicates increased anisotropy. Figure 6c shows that $A_B>A_G$ for the orthorhombic structure, whereas the opposite behavior is observed in the high-pressure phases. The results show that $A_G$ and $A_B$ for scheelite are very small at the transition pressure and decrease with increasing pressure, whereas $A_G$ for zircon is much higher than for scheelite. These results arise from differences between the bulk and shear moduli obtained using the Voigt and Reuss methods.

According to the literature, $A^U$ generalizes the Zener anisotropy index to account for contributions from shear and bulk simultaneously [53]. Results in Figure 6c show behavior similar to that in Figure 6b, with $A^U$ increasing with pressure in the CrVO₄-type phase. After the phase transition, the high-pressure phases follow different trends: $A^U$ is smaller in scheelite and decreases with pressure, whereas it is larger in zircon and increases with pressure. Thus, the $A^U$ results also suggest that InPO₄ could not be stable in the zircon phase at high pressures. Therefore, the high-pressure phase of InPO₄ is most likely the scheelite.

4. Conclusions

We have performed first-principles calculations to investigate the structural parameters and mechanical properties of InPO₄ polymorphs under pressure. Our results consistently indicate that the phase coexistence observed at high pressures in experiments on InPO₄ is unlikely. From our perspective, this may result from one of two factors, or a combination of both: (i) the findings with the DAC might be influenced by the sample itself, the pressure intervals during measurements, or non-hydrostatic conditions caused by the pressure-transmitting medium; and ($ii$) the existence of kinetic barriers that hinder phase transitions, which can only be overcome at high pressures and temperatures. Meanwhile, our results for elastic and mechanical properties, as well as elastic anisotropy, reinforce previously published results on the dynamic stability of InPO₄ under pressure, which indicate a higher probability that the high-pressure phase of InPO₄ is scheelite rather than zircon.

We hope that these results will motivate the scientific community to conduct further experimental studies of InPO₄ under pressure, broadening the discussion of this compound and helping elucidate the behavior of orthorhombic phosphates under pressure, much of which remains to be discovered.