High-Pressure Elastic, Vibrational and Structural Study of Monazite-Type GdPO₄ from Ab Initio Simulations

Abstract

The GdPO₄ monazite-type has been studied under high pressure by first principles calculations in the framework of density functional theory. This study focuses on the structural, dynamical, and elastic properties of this material. Information about the structure and its evolution under pressure, the equation of state, and its compressibility are reported. The evolution of the Raman and Infrared frequencies, as well as their pressure coefficients are also presented. Finally, the study of the elastic constants provides information related with the elastic and mechanical properties of this compound. From our results, we conclude that monazite-type GdPO₄ becomes mechanically unstable at 54 GPa; no evidence of soft phonons has been found up to this pressure at the zone center.

1. Introduction

The APO₄ orthophosphates compounds (A = trivalent cation) are materials analogous to orthoarsenates, orthovanadates, and orthosilicates. The size of the ionic radio of the A cation defines the two different structures where this family of compounds crystallizes. APO₄ compounds with an ionic radio lower than Gd crystallize in the tetragonal zircon structure (I4₁/amd, space group 141 with Z = 4). All other orthophosphates (A = La to Gd) crystallize into a monoclinic lower symmetry phase, that is, a monazite structure (P2₁/n, space group 14 with Z = 4) [1]. The monazite structure is isostructural to cerium phosphate mineral (monazite). This structure can be viewed as being composed by alternating edge-sharing AO₉ polyhedra, with a structural distortion derived from a rotation of the PO₄ tetrahedra and a lateral shift of the (100) planes that reduce the symmetry from the tetragonal of the zircon structure to the monoclinic symmetry.

Due to their numerous potential applications, the orthophosphate family has attracted a lot of interest in the last decades. The study of the mechanical properties of these materials can be relevant, since some orthophosphates have shown to be promising materials for oxidation-resistant ceramic toughening [2] and to have interesting luminescent and optical properties with a wide range of potential applications [3,4,5,6].

This family of minerals appears in nature as accessory mineral in rhyolites and granitoid rocks [1]. Therefore, the study of orthophosphates may be of importance in petrology [7], chemistry, and mineral physics. These materials can be employed as geochronometers for geological dating [8], because of their doping with radioactive elements that possess a very large half-life. Additionally, since orthophosphates incorporate rare earth elements, they can also be used to determine the distribution of rare earth elements in our Planet [9]. Another interesting application of these compounds is related to their use as potential matrix for nuclear waste disposal [5,10,11,12]. GdPO₄ and other orthophosphates have been suggested as inclusion compounds to made nuclear fuel canisters due to their high neutron absorption [13]. Therefore, the study of these orthophosphates under high pressure is very promising for future practical applications. Although there are several studies focused on pressure-induced transition and on the structural and vibrational properties of monazite phosphates under pressure [7,14,15,16,17,18], these studies were carried out only up to 30 GPa. A systematic understanding of their properties has not yet been achieved; the study of their elastic and mechanical properties is scarce [5,19,20]. Clearly, additional researches are needed to deepen our knowledge of this group of materials. A systematic investigation of the properties of monazite phosphates (particularly at high pressure) is necessary, especially if we want to use them in efficient and safe applications.

The ab initio methods have been successfully applied to study the thermodynamic properties of monazite rare earth orthophosphates [21,22,23,24]. Moreover, the use of ab initio simulations has become a strong and very well supported method to investigate the properties of materials under high pressures [25,26,27,28]. In particular, in the case of monazite-type orthophosphates under pressure, the ab initio simulations have been employed with good results [29,30].

The aim of the present paper is to contribute to the knowledge of the properties of orthophosphates at high pressure. In this work, we perform a detailed study of the structural, dynamic, and elastic properties of monazite-type GdPO₄ under pressures up to 60 GPa, going from first principles simulations. This method has proven to be quite efficient, in order to study a variety of properties on related compounds.

2. Simulations Details

The influence of pressure on the crystal structure, mechanical and vibrational properties of monazite-type GdPO₄ has been analyzed using ab initio simulations. The study was based on the Density Functional Theory (DFT) [31] employing the Vienna Ab initio Simulation Package (VASP) [32,33,34] and pseudopotentials generated with the Projector-Augmented Wave scheme (PAW) [35,36], with the f electrons included into the core. This has proven to give good results in the study of monazite-type structures [20,30,37]. To reach accurate results, the set of plane waves was extended up to a 520-eV cutoff energy and the exchange-correlation energy was expressed by means of the Generalized-Gradient Approximation (GGA) with the Perdew-Burke-Ernzerhof for solids (PBEsol) functional [38]. A dense grid of Monkhorst-Pack [39] k-special points was utilized to perform the integrations on the Brillouin Zone (BZ). The achieved convergence was 1 meV per formula unit in the total energy. All the structural parameters were optimized by minimizing the forces on the atoms and the stress tensor, at selected volumes. This method has been successfully applied to study the phase stability, and structural properties of semiconductors under high pressure [40]. To carry out the study of the mechanical properties of GdPO₄, we have evaluated the elastic constants, which describe the mechanical properties of a material in the region of small deformations. The elastic constants can be obtained by computing the macroscopic stress of a small strain with the use of the stress theorem [41,42]. In the present study, we employ the method implemented in the VASP code: the ground state and fully optimized structures were strained in different directions taking into account their symmetry [43].

The phonons study was performed using the supercell method [44]. The diagonalization of the dynamical matrix provides the frequencies of the normal modes. Furthermore, these calculations also provide the symmetry and eigenvectors of the phonon modes at the Γ point. The calculation of the phonon dispersion curves along the main directions within the BZ was carried out with a supercell of size 2 × 2 × 2. Following the conclusions of our previous studies and those of Blanca-Romero et al. [45], in all the calculations, we neglect the spin–orbit interaction, because the structural properties are barely affected [15,17,26,29,30,37].

3. Results and Discussion

3.1. Structural Properties

The monoclinic structure of monazite, space group P2₁/n [1] has been described in the introduction. A schematic view of the crystal structure is presented in Figure 1. It can be seen as an alternating chain of phosphorus-oxygen PO₄ tetrahedra and trivalent cation-oxygen GdO₉ polyhedra.

The calculated structural parameters and atomic coordinates at ambient pressure are given in

Table 1. Structural parameters of monazite-type GdPO₄ at 0 GPa.

	This Study	Experiments	Theory
a (Å)	6.6276	6.623 a 6.6516(3) b 6.62(2) c	6.4152 d 6.713 e
b (Å)	6.8145	6.829 a 6.84840(7) b 6.823(2) c	6.6103 d 6.887 e
c (Å)	6.2930	6.335 a 6.33571(12) b 6.319(2) c	6.0953 d 6.358 e
β (°)	104.18°	103.80 a 104.023(2) b 104.16(2) c	104.6 d 104.2 e

^a Reference [14]. ^b Reference [18]. ^c Reference [11]. ^d Reference [20]. ^e Reference [21]. Data from References [18,20] were obtained in a different setting than the one employed in the present work.

and

Table 2. Atomic positions of monazite-type GdPO₄ at 0 GPa.

Atom	Site	x	y	z
Gd	4e	0.28394	0.14513	0.08817
P	4e	0.29933	0.16215	0.61291
O1	4e	0.2510	0.0000	0.4330
O2	4e	0.3850	0.3381	0.5017
O3	4e	0.4735	0.1010	0.8184
O4	4e	0.1182	0.2100	0.7148

, and are in a very close agreement with the available theoretical and experimental results [11,14,18,20,21]. Previous theoretical results [20] differ slightly due to the exchange-correlation functional used in the calculations.

The evolution of the structural parameters with pressure is displayed in Figure 2a. The calculations have been already compared with experiments [14,18], showing an excellent agreement. The obtained linear axial compressibilities of each axis are: $K_a= \frac{-1}{a} \frac{da}{dp}=2.76\times {10}^{-3} {\mathrm{GPa}}^{-1}; K_b= \frac{-1}{b} \frac{db}{dp}=2.25\times {10}^{-3} {\mathrm{GPa}}^{-1}; K_c= \frac{-1}{c} \frac{dc}{dp}=2.03\times {10}^{-3} {\mathrm{GPa}}^{-1}$. Therefore, from the present results and previous studies [14,18], it can be concluded that the compression of GdPO₄, as in other monazite-type phosphates, is not isotropic. The a-axis is the most compressible one and the c-axis the least compressible one. As shown in Figure 2a, there is a tendency for the unit-cell parameter a to approach the value of c. Moreover, in Figure 2b, it can be seen that the monoclinic β angle decreases under compression. As a result of these two facts, there is a gradual symmetrization of the monazite structure under compression.

The bulk modulus, B₀, and its first pressure derivative were obtained by fitting the set of theoretical energy-volume data with a third-order Birch-Murnaghan (BM) Equation of State (EOS) [46]. The results (B₀ = 138.3 GPa, and B′₀ = 4.02) are summarized in

Table 3. Volume at 0 GPa, bulk modulus, B₀, and its pressure derivative, B′₀, from a fit with a 3th order Birch Murnaghan EOS.

	This Work	Experiments	Theory
V (Å3)	275.55	279.1 a 280.008(4) b 276.4(4) c	250.20 e 284.96 f
B0 (GPa)	138.3	160 a 128.1(8) b 137 d	149 g 121.0 h
B′0	4.07	5.8(2) b

^a Reference [14]. ^b Reference [18]. ^c Reference [11]. ^d Reference [47]. ^e Reference [20]. ^f Refence [21]. ^g Reference [48], ^h Reference [24].

and are compared with previous results of B₀ reported for GdPO₄: the agreement between experiments is good.

From our calculations, we have determined the pressure dependence of the polyhedral volumes and distortions for monazite GdPO₄. Figure 3a presents the relative volume of the PO₄ and GdO₉ polyhedra and that of the unit–cell. As it can be observed, the PO₄ tetrahedron is highly incompressible, while the GdO₉ polyhedron is much more compressible. Actually, in the monazite-type GdPO₄, the volume change of the GO₉ polyhedra seems to be responsible for most of the volume reduction induced by pressure within the structure. In fact, the phosphate tetrahedra can be treated as a rigid unit in the monazite-type phosphate [17,18,48]. The P–O distances do not vary greatly within the tetrahedra. At zero pressure, two distances are 1.55 Å, and the two other 1.56 Å, and 1.54 Å, while at 53 GPa these distances decrease to 1.51 Å × 2, 1.52 Å, and 1.49 Å, respectively. Indeed, the distortion index, calculated with VESTA [49], for the PO₄ polyhedron (see Figure 3b) is very small, and it increases slowly with compression. Regarding the GdO₉ polyhedra at zero pressure, the central gadolinium atom is connected to nine oxygen atoms with eight bonds whose lengths vary from 2.34 to 2.55 Å, with one length at 2.78 Å. At 53 GPa the lengths range from 2.16 to 2.46 Å and the largest one changes to 2.53 Å. The GdO₉ distortion index do not vary linearly, as seen in Figure 3b. There is a decrease around 4 GPa, followed by a considerable increase in pressure, in a similar manner than in the isostructural LaPO₄ and CePO₄. The GdO₉ polyhedron is more distorted than the PO₄ tetrahedron: its distortion index changes from approximately 0.0381 at ambient pressure to 0.0440 at 54 GPa.

3.2. Elastic Properties

GdPO₄ belongs to the monoclinic space group n° 14; it therefore has 13 independent elastic constants [50]: C₁₁, C₁₂, C₁₃, C₁₅, C₂₂, C₂₃, C₂₅, C₃₃, C₃₅, C₄₄, C₄₆, C₅₅, and C₆₆ in the Voigt notation.

Table 4. Elastic constants (in GPa) of monazite-type GdPO₄ at 0 GPa.

C11	C22	C33	C44	C55	C66	C12	C13	C15	C23	C25	C25	C46
262.19	251.61	205.01	65.19	60.32	58.16	71.41	90.99	5.28	88.50	−18.17	−18.17	−13.70

summarizes our results for the set of C_ij elastic constants at 0 GPa. The values here reported are in overall good agreement with those of Feng et al. [20]. The diagonal elastic constants C₁₁, C₂₂, C₃₃, are the highest ones, while C₂₅, C₃₅, C₄₆ are the lowest ones. It must be noted that in [20], the cell parameters are underestimated, as commented above. When a non-zero uniform stress is applied to a crystal, the relevant magnitudes that describe their elastic properties are the elastic stiffness coefficients B_ij [51]. In the special case of a hydrostatic pressure, P, applied to a monoclinic crystal, the elastic stiffness coefficients are related to the elastic constants through the following relationships [51,52]:

B₁₁ = C₁₁ − P, B₁₂ = C₁₂ + P, B₁₃ = C₁₃ + P, B₁₅ = C₁₅, B₂₂ = C₂₂ − P, B₂₃ = C₂₃ + P, B₂₅ = C₂₅, B₃₃ = C₃₃ − P, B₃₅ = C₃₅, B₄₄ = C₄₄ − P, B₄₆ = C₄₆, B₅₅ = C₅₅ − P and B₆₆ = C₆₆ − P

.

Figure 4 shows the pressure dependence of the elastic stiffness coefficients, B_ij. It can be seen that, B₁₁, B₂₂, B₃₃, B₁₂, B₁₃, and B₂₃ increase with pressure in the whole pressure range studied, especially in the diagonal components, while B₄₄, B₅₅ and B₆₆ decrease under compression. On the other hand, B₁₅ is almost unaffected by pressure. It should be noted that B₂₅, B₃₅, B₄₆ have similar negatives values and that they decrease under compression.

At zero pressure, a crystal is mechanically stable when the Born stability criteria are fulfilled [53]. However, when a hydrostatic pressure is applied to the crystal, the generalized Born stability criteria [52,54] must be employed to study the mechanical stability. This means that the matrix B_ij must be positive definite. Consequently, the stability conditions for a monoclinic crystal are given by the following conditions:

$$M_1=B_{11}>0$$

$$\begin{array}{c}{M}_2=B_{11}{B}_{22}-B_{12}^2>0\\ M_3=(B_{22}{B}_{33}-B_{23}^2)B_{11}-B_{33}{B}_{12}^2+2B_{23}{B}_{12}{B}_{13}-B_{22}{B}_{13}^2>0\end{array}$$

$$M_4=B_{44}>0$$

$$\begin{array}{l}{M}_5=B_{12}^2{B}_{35}^2-B_{33}{B}_{55}{B}_{12}^2+2B_{55}{B}_{12}{B}_{13}{B}_{23}-2B_{12}{B}_{13}{B}_{25}{B}_{35}\\ -2B_{12}{B}_{15}{B}_{23}{B}_{35}+2B_{33}{B}_{12}{B}_{15}{B}_{25}+B_{13}^2{B}_{25}^2-B_{22}{B}_{55}{B}_{13}^2\\ -2B_{13}{B}_{15}{B}_{23}{B}_{25}+2B_{22}{B}_{13}{B}_{15}{B}_{35}+B_{15}^2{B}_{23}^2-B_{22}{B}_{33}{B}_{15}^2\\ -B_{11}{B}_{55}{B}_{23}^2+2B_{11}{B}_{23}{B}_{25}{B}_{35}-B_{11}{B}_{33}{B}_{25}^2-B_{11}{B}_{22}{B}_{35}^2\\ +B_{11}{B}_{22}{B}_{33}{B}_{55}>0\end{array}$$

$$M_6=B_{44}{B}_{66}-B_{46}^2>0$$

The simulations of the elastic constants have been extended up to 60 GPa, well above the maximum experimental pressure reported [14,15]. The generalized stability criteria for GdPO₄ as function of the pressure are presented in Figure 5. At 54 GPa, one of the M₆ criterion is violated, making the system mechanically unstable at this pressure. No evidence of soft phonons was found at the Γ point in this pressure range. These results are consistent with the experimental and theoretical reports that confirm that there is no phase transition induced by pressure in the monazite structure up to the highest pressure reached in those studies (30 GPa) [14,17].

The bulk modulus (B), the shear modulus (G), the Young modulus (E), and the Poisson’s ratio (ν) describe the major elastic properties of a material. Analytical expressions can be obtained for these moduli in the Voigt, Reuss and Hill approximations [55,56,57] from the elastic stiffness coefficients, B_ij. Hill has proved that the Voigt and Reuss approximations are limits and pointed out that the arithmetic mean of the two bounds can be considered as the actual B and G elastic moduli. In the case of a monoclinic crystal, the bulk and shear moduli can be expressed in the three approximations as [58]:

$$B_V=\frac{B_{11}+B_{22}+B_{33}+2(B_{12}+B_{23}+B_{13})}{9}$$

$$\frac{1}{B_R}=S_{11}+S_{22}+S_{33}+2(S_{12}+S_{23}+S_{13})$$

$$B_H=\frac{B_V+B_R}{2}$$

$$G_V=\frac{B_{11}+B_{22}+B_{33}-(B_{12}+B_{23}+B_{13})+3(B_{44}+B_{55}+B_{66})}{15}$$

$$\frac{1}{G_R}=\frac{4(S_{11}+S_{22}+S_{33})-4(S_{12}+S_{23}+S_{13})+3(S_{44}+S_{55}+S_{66})}{15}$$

$$G_H=\frac{G_V+G_R}{2}$$

where S_ij refers to components of the elastic compliances tensor, and subscripts V, R, and H stand for Voigt, Reuss and Hill respectively. The Young modulus, E, and the Poisson’s ratio, ν, are obtained with the expressions [59]:

$$E_X=\frac{9B_X{G}_X}{G_X+3B_X}$$

$${\nu }_X=\frac{1}{2}\left(\frac{3B_X-2G_X}{3B_X+G_X}\right)$$

where the subscript X refers to the symbols V, R, and H.

In order to discuss the elastic properties of GdPO₄ at ambient pressure, the elastic moduli at 0 GPa are summarized in

Table 5. Elastic moduli B, E and G, the B/G ratio, and the Poisson’s ratio, ν, in the Hill approximation, at 0 GPa.

BH (GPa)	EH (GPa)	GH (GPa)	BH/GH	ν
134.47 a	169.62 a	65.575 a	2.05 a	0.289 a
137 b	172 b	67 b	2.045 b	0.290 b
121.0 c	165.2 c	64.9 c	1.86 c

^a This work, ^b Reference [47], ^c Reference [24].

. The bulk modulus, B_H = 134.47 GPa is in rather good agreement with our theoretical value of B₀ = 138.3 GPa, obtained from a fit with a third order Birch-Murnaghan equation of state [46], and with the experimental values of B₀ = 137 GPa previously reported [47]. As for E, G, and B/G elastic moduli, as well as the Poisson’s ratio, ν, at zero applied pressure, it should be noted once more that there is good agreement between our calculations and the experimental results. These facts demonstrate the quality of our simulations.

The pressure dependence of the elastic moduli up to 56.1 GPa is presented in Figure 6a–d. It can be observed that B_H, and ν_H increase with pressure. However, E_H and G_H increase under compression reaching maximum values at 11 GPa and 9 GPa, respectively; above that pressure they decrease. All the elastic moduli change dramatically around 54 GPa, since the monazite structure becomes mechanically unstable. The Poisson’s ratio gives information about the characteristics of the bonding forces and the chemical bonding. The value of the Poisson’s ratio in the Hill approximation is ν_H = 0.29 at 0 GPa. A value of ν > 0.25 indicates that the interatomic bonding forces are predominantly central and that ionic bonding is predominant against covalent bonding [60,61]. The increase of ν upon pressure can be interpreted as an increase of the metallization.

The B/G ratio is 2.05 and it increases with pressure (see Figure 7). Therefore, GdPO₄ monazite is more resistant to volume compression than to shear deformation (B > G). Moreover, according to Pugh criterion, monazite-type GdPO₄ behaves like a ductile material, since B/G is smaller than 1.75 [62], in the whole pressure range studied (materials with B/G < 1.75 behave as brittle materials). Our B/G value is consistent with the experimental result of Du et al. [47], although it is somewhat larger than the theoretical one of reference [24]. This is probably due to the small value of B₀ reported in that reference. However, it is clear that monazite GdPO₄ is a ductile material.

3.3. Vibrational Properties

The phonon dispersion of GdPO₄ along the main paths on the Brillouin zone is presented in Figure 8. Three frequency regions are separated by two gaps: a little one between the low- and the medium-frequency regions, and a bigger one between the medium- and the high-frequencies regions. The Partial Density Of States (PDOS) for phonons (see Figure 9) allows us to analyze these vibrations. The PDOS presents three zones separated by two gaps. The first zone, corresponding to the low-frequencies, down to 311 cm⁻¹, is composed of vibrations of Gd atoms and GdO₉ polyhedra, as well as a small contribution of PbO₄ tetrahedra. The intermediate-frequency zone, from 381 cm⁻¹ to 600 cm⁻¹, is mainly due to vibrations of the PbO₄ tetrahedra with a very small contribution from the GdO₉ polyhedra. A large gap of ≈340 cm⁻¹ separates this zone with the high-frequency region, which corresponds to frequencies between 942 cm⁻¹ and 1073 cm⁻¹ that are only due to vibrations from PbO₄ units. Therefore, the vibrational spectra of the GdPO₄ monazite could be interpreted in term of the modes of the PbO₄ tetrahedra, which can be considered as independent units in the monazite structure, and GdO₉ polyhedra. Indeed, the vibrational spectrum of monazites ABO₄ has been interpreted in terms of internal modes (bending and stretching modes) associated with the tetrahedron BO₄ and external modes (translational and rotational) which involves movement of both A and BO₄ ions [17,63].

Through group theory analysis, it can be established that the monazite structure has 72 vibrational modes at the zone center: 36 optical Raman-active modes (18A_g + 18B_g), 33 optical IR-active modes (17A_u + 16B_u), and 3 acoustic modes (1A_u + 2B_u). These vibrational modes can be interpreted as 36 internal (ν₁, ν₂, ν₃ and ν₄) and 36 external (translational (T) and rotational (R)) modes [17,63].

The Raman spectrum of many monazite-type phosphates (e.g., BiPO₄ LaPO₄, CePO₄, PrPO₄ and GdPO₄) at ambient conditions has been previously studied by different authors [18,29,64,65]. BiPO₄ LaPO₄, CePO₄, and PrPO₄ have also been studied under compression [17,66,67,68]. The frequency distribution of Raman modes is very similar for all of them. As in the case of GdPO₄, phonon gaps are observed between the three frequency regions in monazite phosphates; they are also observed in the monazite structure of chromates and selenates [30].

GdPO₄ has a distribution of Raman modes similar to other monazite phosphates [17]. The Raman mode frequencies and their pressure coefficients at room pressure are summarized in

Table 6. Raman frequencies at ambient pressure, their pressure coefficients, and Grüneisen parameters (γ). The pressure dependence of frequency has been fitted with a second order polynomial: ${\mathsf{\omega }=\omega }_0+\left(\frac{\partial \omega }{\partial P}\right)\mathrm{P}+\left(\frac{{\partial }^2\omega }{\partial P^2}\right)P^2$.

Mode	${\mathsf{\omega }}_0$ (cm−1)	$\frac{\partial \mathit{\omega }}{\partial \mathbf{P}}$ (cm−1/GPa)	$\frac{{\partial }^2\mathbf{\omega }}{\partial {\mathbf{P}}^2}$ (cm−1/GPa2)	γ
Bg	84.78	0.762	0.0180	1.24
Ag	86.33	0.807	0.0128	1.29
Ag	108.61	0.846	0.0216	1.07
Ag	124.99	0.765	0.008	0.84
Bg	129.32	0.830	0.016	0.88
Bg	137.84	0.852	0.002	0.85
Bg	153.93	0.059	0.027	0.05
Ag	178.30	1.354	0.029	1.05
Bg	180.54	2.610	0.007	1.99
Ag	190.93	1.648	0.016	1.19
Ag	189.67	3.477	0.0343	2.53
Bg	235.29	3.887	0.057	2.28
Ag	240.14	3.014	0.0233	1.736
Bg	246.86	3.411	0.012	1.911
Ag	264.93	5.089	0.069	2.656
Bg	276.56	3.176	0.011	1.588
Bg	295.93	3.982	0.044	1.861
Ag	310.90	2.209	0.000	0.982
Bg	381.29	2.722	0.001	0.987
Ag	404.71	2.524	0.007	0.862
Ag	455.84	2.592	0.018	0.786
Bg	499.31	2.653	0.035	0.734
Ag	506.02	0.589	0.010	0.161
Bg	533.07	1.190	0.017	0.308
Ag	542.17	1.677	0.003	0.427
Bg	563.55	1.750	0.004	0.429
Ag	600.11	1.482	0.007	0.341
Bg	603.45	1.624	0.007	0.372
Bg	942.72	4.215	0.028	0.618
Ag	954.05	4.277	0.036	0.620
A	984.01	4.300	0.013	0.604
Ag	1007.23	5.005	0.039	0.687
Bg	1020.45	3.972	0.016	0.538
A	1051.30	4.269	0.014	0.561
Bg	1055.21	4.837	0.050	0.633
Bg	1072.33	4.480	0.008	0.577

. The mode Grüneisen parameters [69], that provide a dimensionless representation of the response to compression are also reported.

It can be observed that the Raman spectrum of GdPO₄ monazite could be divided into three regions: the low-frequency region, up to 310.91 cm⁻¹, corresponds to the eighteen external or lattice T and R modes (9A_g + 9B_g); the medium-frequency region, between 381.30 and 603.46 cm⁻¹, corresponds to the ten internal bending modes (5A_g + 5B_g); and the high-frequency region, above 942.72 cm⁻¹, corresponds to the eight internal stretching modes (4A_g + 4B_g).

As seen in Figure 10, which shows the theoretical pressure dependence of the Raman-active mode frequencies, most modes harden under compression, as is usual in most materials. However, there are a few modes between 84.78 and 153.94 cm⁻¹ whose frequency changes very slightly under compression.

External modes (low frequency region) involve movements of the trivalent Gd atoms, and in phosphate monazites ABO₄, their frequencies greatly depend on the mass of the A atom [17,70]. These external modes show very different pressure coefficients, since they involve different Gd-O bonds, with different compressibility, as seen in Section 2 [14]. Due to the marked differences between the pressure dependence of the different modes, see Table 6, crossing and anti-crossing phenomena are observed in this region. In particular, one of the modes most affected by compression is the A_g mode at 264.93 cm⁻¹; The lowest-frequency B_g mode is almost not affected by pressure. In previous studies of monazite chromates [37] and phosphates [17], softening of the lower Raman modes was related to a pressure-driven instability of the monazite structure, which could result in a phase transition above 30 GPa [17]. In the case of the monazite-type GdPO₄, the lower modes are little affected by compression. As commented in a previous section, the extension of the simulation up to 60 GPa does not show evidence of soft phonons mode at the Γ point. The pressure-driven instability seems to be related to mechanical instability, rather than dynamical instability.

Internal bending motions of the PO₄ tetrahedron (medium frequencies region) have smaller pressure coefficients. Two modes with frequencies A_g = 506.02 and B_g = 530 cm⁻¹ at room pressure are the less affected by pressure, while the mode most sensitive to pressure in this region is the B_g mode with a frequency 381.30 cm⁻¹ at room pressure. The crossover of B_g (499.32 cm⁻¹) and A_g (506.02 cm⁻¹) modes is observed at ≈4 GPa in Figure 10, due to the different pressure dependences. This behavior could be related to the non-isotropic compression of monazite. This can cause the lower-frequency B_g mode to move faster towards high frequencies than the higher-frequency mode.

The internal stretching modes (high frequency region) of the PO₄ tetrahedron have similar pressure coefficients, and they are among the modes whose frequency increases faster under compression.

Regarding the infrared modes of monazite GdPO₄, low-, medium-, and high frequencies modes have the same general behavior with pressure than the Raman modes have. This also happens for monazite chromates [34]. In particular, the modes less affected by pressure is the first A_u mode (91.057 cm⁻¹); see

Table 7. Infrared frequencies at ambient pressure, their pressure coefficients, and Grüneisen parameters (γ). The pressure dependence of frequency has been fitted with a second order polynomial: ${\mathsf{\omega }=\omega }_0+\left(\frac{\partial \omega }{\partial P}\right)\mathrm{P}+\left(\frac{{\partial }^2\omega }{\partial P^2}\right)P^2$.

Mode	${\mathsf{\omega }}_0$ (cm−1)	$\frac{\partial \mathit{\omega }}{\partial \mathbf{P}}$ (cm−1/GPa)	$\frac{{\partial }^2\mathbf{\omega }}{\partial {\mathbf{P}}^2}$ (cm−1/GPa2)	γ
Au	91.05	0.219	0.007	0.332
Bu	102.50	1.442	0.005	1.946
Au	116.32	1.541	0.0125	1.832
Bu	168.18	0.722	0.013	0.594
Au	170.55	0.866	0.010	0.702
Bu	187.21	1.270	0.032	0.938
Au	189.765	2.761	0.041	2.012
Bu	205.99	2.328	0.021	1.563
Au	219.28	1.386	0.006	0.874
Bu	225.70	3.603	0.026	2.207
Au	245.96	3.397	0.032	1.910
Bu	254.49	3.153	0.013	1.713
Au	274.99	3.789	0.0231	1.905
Au	304.38	4.034	0.039	1.832
Bu	309.61	1.867	0.006	0.834
Bu	375.28	3.061	0.033	1.128
Au	388.32	2.186	0.0011	0.778
Au	466.32	1.013	0.0122	0.300
Bu	484.23	1.851	0.003	0.528
Au	507.73	1.501	0.0044	0.408
Bu	526.89	2.3165	0.0124	0.608
Au	533.09	1.864	0.0033	0.483
Bu	550.13	1.428	0.007	0.359
Bu	590.07	1.384	0.006	0.324
Au	613.26	1.765	0.008	0.398
Au	935.32	4.083	0.023	0.603
Bu	936.86	4.205	0.029	0.620
Au	974.23	3.643	0.017	0.517
Bu	986.50	3.483	0.014	0.488
Bu	1004.66	4.674	0.030	0.643
Au	1014.56	5.209	0.038	0.710
Bu	1066.28	4.956	0.024	0.642
Au	1082.61	4.763	0.024	0.608

and Figure 11.

The Grüneisen parameters in the high frequency region are very similar, while there are large differences between the Grüneisen parameters for the lower frequencies (Table 6 and Table 7). Therefore, there are large differences between the restoring force on the atoms associated with the lowest modes.

4. Conclusions

The effects of high pressure on the structure of monazite-type GdPO₄ were studied from first principles, ab initio calculations. The simulations were extended up to 60 GPa, well above the maximum experimental pressure previously reported (30 GPa). The evolution of the lattice parameters under compression, the equations of state, and the axial and polyhedral compressibilities are reported. Moreover, the under-pressure distortion of the polyhedral units PO₄ and GdO₉ is shown. The compression of monazite-type GdPO₄ is not isotropic, and the GdO₉ polyhedra account for almost all the volume reduction under pressure.

The elastic constants and elastic stiffness coefficients were accurately determined, through which the elastic behavior of monazite-type GdPO₄ at high pressure was analyzed. The evolution with pressure of the B, G, and E elastic moduli, ν Poisson’s ratio, and the B/G ratio was reported. At all pressures, this compound has shown to be ductile and more resistive to volume compression than to shear deformation (B > G). Furthermore, the high-pressure mechanical stability of this monoclinic compound was studied. It was found that monazite-type GdPO₄ becomes mechanically unstable above 54 GPa.

Finally, through the total and partial density of states, the contribution of each polynomial units to the vibrational modes was discussed. This contribution is similar to that found in other monazite phosphates. The evolution of the Raman- and infrared active modes of the monazite-type GdPO₄ as function of pressure was analyzed. No soft-modes were found up to 60 GPa at the center of the Brillouin zone.

A possible pressure-driven phase transition or amorphization might occur due to mechanical instability at the high pressure of 54 GPa. No dynamical instability was found up to this pressure.

The results of the presents study confirm the high stability under pressure of the monazite-type GdPO₄, which is very promising for practical applications.