Pressure-Induced Monoclinic to Tetragonal Phase Transition in RTaO₄ (R = Nd, Sm): DFT-Based First Principles Studies

Abstract

In this manuscript, we report the density functional theory-based first principles study of the structural and vibrational properties of technologically relevant M′ fergusonite (P2/c)-structured NdTaO₄ and SmTaO₄ under compression. For NdTaO₄ and SmTaO_4, ambient unit cell parameters, along with constituent polyhedral volume and bond lengths, have been compared with earlier reported parameters for EuTaO₄ and GdTaO₄ for a better understanding of the role of lanthanide radii on the primitive unit cell. For both the compounds, our calculations show the presence of first-order monoclinic to tetragonal phase transition accompanied by nearly a 1.3% volume collapse and an increase in oxygen coordination around the tantalum (Ta) cation from ambient six to eight at phase transition. A lower bulk modulus obtained in the high-pressure tetragonal phase when compared to the ambient monoclinic phase is indicative of the more compressible unit cell under pressure. Phonon modes are calculated for the ambient and high-pressure phases with compression for both the compounds along with their pressure coefficients. One particular IR mode has been observed to show red shift in the ambient monoclinic phase, possibly leading to the instability in the compounds under compression.

1. Introduction

RBO₄ (R: rare earth; B: a pentavalent cation such as V, W, Mo, Nb, Ta, As, and P) compounds are the subject of extensive research due to their promising applications in areas such as proton-conducting solid oxide fuel cells [1], and as a host for nuclear radioactive waste immobilization [2,3]. It has been well established that the B cation plays a crucial role in deciding the stable structure of RBO₄ compounds. Depending on the ratio of B cationic radii to lanthanide radii, RBO₄ compounds are reported to crystallize in either tetragonal (zircon, scheelite type) or monoclinic (fergusonite, monazite, wulframite type) structures [4]. Rare earth orthovanadate (RVO₄) are generally synthesized in a zircon (tetragonal) structure with a I4₁/amd space group, while rare earth orthotungstates (RWO₄) and molybates (RMoO₄) have been reported to crystallize in scheelite (tetragonal) structure with I4₁/a crystal symmetry [5,6,7]. A monoclinic M fergusonite structure with space group I2/a has been established as a stable structure for rare earth orthoniobates (RNbO₄) at ambient temperature and pressure [8]. Depending upon the atomic radii of the lanthanide cation, the crystal structure of rare earth orthophosphate is either zircon (R ˂ Gd) or monazite (R ≥ Gd) [9]. All the structures are closely related to each other by group–subgroup relations. A tetragonal scheelite structure (I4₁/a) is a subgroup of a tetragonal zircon (I4₁/amd) structure and the transformation between these two structures is generally of the first-order reconstructive type. A scheelite structure is transformed to a fergusonite structure by means of another translationgleiche, which involves lowering of point group symmetry from 4/m to 2/m [10]. Among all RBO₄ compounds, the RTaO₄ family of compounds exhibit polymorphism at ambient pressure and temperature conditions, which makes them of great interest from theoretical and technological point of views [11,12]. In RTaO₄ compounds, the final stable structure not only depends on the lanthanide radii but also depends on the heating temperature. Most of the compounds belonging to the RTaO₄ family stabilize either in a M fergusonite or M’ fergusonite structure at room temperature depending on the processing parameters. The main difference between the M fergusonite and M’ fergusonite structures is the oxygen coordination around the Ta atom. In a M fergusonite structure, the Ta atom is surrounded by an oxygen tetrahedra, whereas an oxygen octahedra is formed in a M’ fergusonite structure. In the M fergusonite phase, doubling of the b axis has been observed compared to M′ structure, while the other two unit cell axes have almost the same value in both structures. In recent times, RTaO₄ family compounds are being investigated extensively due to their potential applications in the field of scintillators and laser materials, owing to their high thermal stability and good chemical stability [13,14]. These compounds are also proposed as excellent alternatives to using Yttria-stabilized zirconia as ceramic thermal barrier coatings (TBCs) due to their lower thermal conductivity and better fracture toughness at high temperatures [15]. As is well known, most often it is the crystal structure that determines the properties of a material, and the structure may be altered by varying the thermodynamic parameters. Recently, by x-ray diffraction, the Raman spectroscopic technique and density functional theory (DFT)-based first principles calculations, we have shown the structural instability of EuTaO₄ and GdTaO₄ [16,17]. In continuation to this work, to understand the compression behavior of the RtaO₄ family of compounds, we have performed density functional theory-based first-principle simulations on technologically important NdTaO₄ and SmTaO₄. Wenhui Xiao et al. [18] have reported that the M’ fergusonite structure is more stable compared to the M fergusonite structure. Therefore, in the present work, we report the results from DFT-based first principles calculations on the M’ fergusonite structure under compression. We have also calculated the equation of state for both the compounds using the third-order Birch–Murnaghan equation of state (BM-EOS). Compressibility analysis of the simulated volume of constituent polyhedral units RO₈ and TaO₆ indicates that the major contribution to the bulk modulus comes from RO₈ polyhedra. This behavior validates Hazen and Finger’s proposed empirical model for predicting the bulk modulus, with contributions from the rare earth polyhedral unit as seen in RVO₄, RWO₄, RMoO_4, RNbO₄ compounds [6,7,19,20,21,22,23,24,25]. We have also calculated the pressure evolution of the Raman and IR modes, which is consistent with earlier reported results for EuTaO₄ and GdTaO₄ for the same structure [16,17]. Further, we have compared the results from this work with previously investigated EuTaO₄ and GdTaO₄ to establish the role of lanthanide contraction present in the RTaO₄ family.

2. Computational Details

DFT-based first-principle simulations were carried out as implemented in Quantum Espresso [26] for determination of stable structures at ambient pressure as well as to investigate the influence of pressure on structural and vibrational properties. This is based on density functional theory, plane waves, and pseudopotentials. The projector-augmented wave (PAW) scheme [27], which describes electron–ion interactions, was employed as pseudo potential in self-consistency field calculation. Appropriate pseudopotentials are taken from the Pslibrary [28] considering 11 valence electrons for Nd (5s²6s^1.55p⁶6p^0.55d¹) and Sm (5s²6s^1.55p⁶6p^0.55d¹), 27 valence electrons for Ta (4f¹⁴ 5d³5p⁶6s²5s²) and 6 valence electrons for oxygen (2s² 2p⁴). A prescribed generalized gradient approximation (GGA) based on the parametrization proposed by Perdew, Burke, and Ernzherhof (PBE) [29] has been accounted for the for calculation of exchange and correlation energy for both the compounds with the lowest-energy M’ fergusonite structure. Wave functions in the Kohn–Sham equation are expanded in a plane wave basis set due to the major advantage of orthonormality and since it is easy to control the convergence with respect to the size of the basis with only one parameter E_cut. In our calculation, the plane wave basis was extended up to 70 Ry for both the compounds (NdTaO₄ and SmTaO₄) to achieve highly converged results in the PAW scheme after thoroughly going through the convergence test. A dense Monkhorst pack grid of 8 × 8 × 8 is used for Brillouin zone integrations. Geometric optimization of NdTaO₄ and SmTaO₄ structures has been achieved using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization algorithm [30], where the structures have been fully optimized to the equilibrium condition at ambient pressure, by minimizing the forces on the atoms and the stress tensor. Helmann–Feynman forces lower than 0.00003 eV/atom on each atom in the unit cell and maximum deviation among the diagonal components of the stress tensor on a unit cell lower than 0.1 GPa ensure a fully relaxed structure. The same steps were followed while calculating from ambient pressure to 50 GPa with an interval of 1 GPa. After obtaining the equilibrium structures at different pressures, phonon frequencies were calculated at the center of the Brillouin zone using density functional perturbation theory (DFPT) as implemented in the Quantum Espresso code [31]. Simulations were performed at zero temperature and under a hydrostatic environment. The stable structures and transition pressures were obtained by analyzing the enthalpy–pressure curve.

3. Results and Discussions

3.1. Ambient Structure

The optimized volume obtained at ambient pressure for both the compounds in the M’ fergusonite structure matches very well with previously reported experimental values [32]. Unit cell parameters obtained from the geometrically relaxed structure with constituent bond lengths along with the previously reported experimental data are given in

Table 1. Comparison of simulated ambient pressure lattice parameters and constituent bond lengths of NdTaO₄ and SmTaO₄ with previously reported experimental values.

Lattice Parameters	NdTaO4		SmTaO4
	Experiment [32]	Calculated	Experiment [32]	Calculated
a (Å)	5.2437(4)	5.257	5.2065(4)	5.206
b (Å)	5.5969(4)	5.638	5.5542(4)	5.571
c (Å)	5.4275(4)	5.451	5.3947(4)	5.397
β (degree)	96.767(9)	96.79	97.721(9)	96.74
Bond length	2×	2×	2×	2×
R-O2 (Å)	2.454	2.459	2.414	2.4179
R-O1(Å)	2.371	2.371	2.346	2.33622
R-O1(Å)	2.608	2.603	2.586	2.57709
R-O2(Å)	2.408	2.409	2.377	2.37666
Ta-O1(Å)	1.864	1.879	1.874	1.88026
Ta-O2(Å)	2.201	2.230	2.211	2.22584
Ta-O2(Å)	1.996	2.003	1.990	1.99934

. The equilibrium volume obtained for NdTaO₄ is 159.1 ų and 155.4 ų for SmTaO₄, which are within 1% and 0.3% of the experimentally observed volume. Figure 1 shows the polyhedral representation of NdTaO₄ and SmTaO₄ belonging to a P2/c (space group no.:13, Z = 2) structure with 2/m point group symmetry. The rare earth cation (Nd/Sm) is surrounded by eight oxygen, forming dodecahedra while oxygen coordination around the Ta cation is six. The formation of oxygen octahedra around tantalum is a distinctive signature of the M’ fergusonite structure that makes it different from the M fergusonite structure with space group I2/a (space group: 15). Each rare earth cation is bonded with eight oxygen with four different R-O bond distances while each Ta cation bonds to six oxygen with three different bond distances, making all the constituent units distorted. These distorted polyhedral units provide structural stability to these compounds against a large range of pressure/temperature, compared to zircon- or scheelite-structured compounds [6,7,19,20,21,22,23,24,25]. In the calculated M’ fergusonite structure, the Nd/Sm and Ta atoms occupy the 2e and 2f Wyckoff positions while oxygen atoms O1 and O2 occupy the 4g position. In Figure 2, we have plotted ambient pressure lattice parameters, the unit cell volume along with the constituent polyhedral volume (RO₈, TaO₆), and R-O and Ta-O bond lengths for both the compounds along with previously reported data for EuTaO₄ and GdTaO₄ for a better understanding of the role of the rare earth cation. An increasing linear trend has been observed in all the parameters with lanthanum cationic radii except the volume of TaO₆ octahedra and its constituent bond lengths. This indicates the major influence of the R cation on structural parameters. In all four RTaO₄ compounds, the polyhedral volume of all distorted TaO₆ octahedra remains almost the same.

3.2. Structural Behavior under Compression

3.2.1. The Low-Pressure Phase

Simulated unit cell parameters for both the compounds, NdTaO₄ and SmTaO₄, have been plotted at different pressures in Figure 3a,b. The anisotropic compressibility of the b axis is clearly seen, which is 2 fold as compressible compared to the other two axes. This particular behavior has been observed in almost all RBO₄ compounds irrespective of their ambient structure [5,6,7,9]. The axial compressibility obtained by fitting the calculated lattice parameters to third-order BM-EOS [33] for NdTaO₄ is K_a = 1.89 × 10⁻³ GPa⁻¹, K_b= 3.79 × 10⁻³ GPa⁻¹ and K_c = 1.75 × 10⁻³ GPa⁻¹; and SmTaO₄ are K_a = 1.66 × 10⁻³ GPa⁻¹, K_b = 3.64 × 10⁻³ GPa⁻¹ and K_c = 1.60 × 10⁻³ GPa⁻¹, respectively. The structural arrangement of relatively more compressible RO₈ polyhedral units along the b axis could be the reason for anisotropic compression as has been reported in other rare earth metal oxides. Similar behavior has also been observed in EuTaO₄ and GdTaO₄ [16,17]. It is interesting to note that a decreasing trend is observed in the axial compressibility as we go from higher ionic radii to lower ionic radii of the lanthanide cation, indicating the more incompressible behavior of RTaO₄ compounds with a lower ionic radii lanthanide cation. Figure 4 shows the pressure evolution of the unit cell volume for both the compounds. The bulk modulus obtained by fitting the simulated pressure–volume data to third-order BM-EOS is 145.1 and 147.2 GPa for NdTaO₄ and SmTaO₄, respectively, which is in good agreement with the bulk modulus reported for the RTaO₄ family of compounds. The bulk modulus reported for NdVO₄ [34] in the zircon and scheelite structures is 124.2 and 136 GPa, respectively, which is similar to the obtained bulk modulus of NdTaO₄ in the present studies. A similar bulk modulus is observed for NdNbO₄ (138.32 GPa) [18]. Similarly, for SmTaO₄, the obtained bulk modulus in the present work is similar to that reported for other SmBO4 compounds [35]. This indicates that it is NdO₈/SmO₈ polyhedral units that mostly contribute to the bulk modulus. A similar bulk modulus for various RBO₄ compounds has already been reported and reaffirms that it is indeed the lower valence polyhedral units (RO₈) that mainly contribute to the bulk modulus [16,17]. This can be validated using Hazen and Finger’s proposed empirical model B₀ = N × Z/(d_R-O)³, where B₀ is the bulk modulus, N is the dimensional-less proportional constant (610 for tantalates and niobates), Z is the formal charge of the R cation and d_R-O is the average cation–anion distance, which only considers the rare earth polyhedral unit for predicting the bulk modulus [36]. For NdTaO₄ with d_Nd-O= 2.460 Å and Z = 3, the calculated bulk modulus using Hazen and Finger’s equation is 122.9 GPa, which is similar to the simulated value obtained in the present work. For the SmTaO₄ compound, the bulk modulus obtained using d_Sm-O = 2.460 Å is 127.4 GPa.

3.2.2. The High-Pressure Phase

Earlier, based on our X-ray diffraction and Raman spectroscopic measurements along with DFT-based first principles calculations on EuTaO₄ and GdTaO₄, an isostructural transition was reported with a sudden drop in the monoclinic angle from 96° to 90° at the transition. This isostructural monoclinic phase could also be described as the pseudo orthorhombic structure due to all unit cell angles of 90°. Therefore, orthorhombic Pcna was tested as an alternative description against the isostructural monoclinic phase and our calculation reveals that, at high pressure, both the structures are energetically favorable. Since NdTaO₄ and SmTaO₄ are found to be synthesized in the same space group (P2/c) as EuTaO₄ and GdTaO₄, the same orthorhombic space group has been tested as the possible high-pressure phase for both the compounds in the present work. For NdTaO_4, the orthorhombic structure with the Pcna space group becomes energetically favorable at approximately 40 GPa as seen in Figure 5a, which depicts enthalpy difference as a function of pressure (the monoclinic phase P2/c has been taken as a reference). Similar behavior has been observed at approximately 33.5 GPa for SmTaO₄ (Figure 5b). This orthorhombic structure can also be alternatively described by a pseudo tetragonal structure due to almost the same value of lattice parameter a and c as seen in calculation. Therefore, the higher-symmetry tetragonal structure P4/nbm with 4/mmm point group symmetry was also tested against the Pcna structure and indeed tetragonal P4/nbm is a lower-energy structure at approximately 33 GPa for SmTaO₄ and approximately 40 GPa for NdTaO₄ as seen in Figure 5a,b. The phase transition in both the compounds is associated with a nearly 1.3% volume collapse at transition pressure along with change in oxygen coordination around the Ta cation from ambient six to eight at phase transition. In Figure 3, we have plotted lattice parameters of the high-pressure tetragonal phase (HP) along with unit cell parameters corresponding to the low-pressure monoclinic phase (LP). The axial compressibility obtained by fitting the calculated lattice parameters to third-order BM-EOS for NdTaO₄ and SmTaO₄ in the HP phases is K_a = 3.75 × 10⁻³ GPa⁻¹, K_c = 1.02 × 10⁻³ GPa⁻¹ and K_a = 3.4 × 10⁻³ GPa⁻¹, K_c = 1.04 × 10⁻³ GPa⁻¹, respectively. Unit cells for both the compounds also undergo anisotropic compression in the high-pressure tetragonal structure as observed in the low-pressure monoclinic structure. The simulated pressure volume data fitted with the third-order Birch–Murnaghan (BM) equation of state (EOS) yields a bulk modulus of 123.74 GPa for NdTaO₄ and 130.60 GPa for SmTaO₄. It is interesting to note that the high-pressure phase has a lower bulk modulus than the low-pressure phase, indicating a more compressible high-pressure phase in spite of volume collapse at phase transition. This can be understood by analyzing the compressional behavior of the constituent polyhedral unit in both the LP phase and the HP phase. In low-pressure phase, the compounds NdTaO₄ and SmTaO₄ are made up of highly compressible NdO₈ and SmO₈ polyhedra as evident from their value of bulk modulus being 117.3 and 115 GPa respectively, while TaO₆ octahedra in both the compounds is highly incompressible with incompressibility modulus being 262.6 and 258.7 GPa for NdTaO₄ and SmTaO₄ respectively. The modulus of incompressibility has been obtained using calculated pressure and polyhedral volume fitted to third-order BM-EOS. A large difference in the bulk modulus at the LP phase affirms that indeed lower valence rare earth polyhedra significantly contribute to the compressibility of the compound in the low-pressure monoclinic phase, which validates Hazen and Finger’s proposed empirical model for predicting the bulk modulus, taking contribution from rare earth polyhedral unit as seen in RVO₄, RWO₄, RMoO₄, RNbO₄ compounds [6,7,19,20,21,22,23,24,25]. In Figure 6a,b, we have plotted the distortion index of bond lengths for the polyhedral units of both NdTaO₄ and SmTaO₄ compounds, which show that RO₈ polyhedra become fully symmetric under compression, while that of TaO₆ more distorted and distortion in the bond length increases sharply at phase transition with the formation of a TaO₈ polyhedral unit. The distortion index of the bond length as defined by Baur [37] has been computed using VESTA software [38]. For the NdTaO₄ compound, the bulk modulus obtained for constituent polyhedral NdO₈ and TaO₈ in the HP phase is 105.9 and 99.8 GPa, respectively, whereas the bulk modulus for the SmTaO₄ compound obtained for constituent polyhedral SmO₈ and TaO₈ in the HP phase is 109.4 and 111.42 GPa, respectively. The similar bulk modulus indicates equal contribution to the compressibility of the high-pressure unit cell unlike the low-pressure phase. Incompressible TaO₆ octahedra change to very compressible TaO₈ at phase transition due to the increase in oxygen coordination under pressure and can be cited as among the reasons for the lower bulk modulus of the high-pressure phase. It should also be noted that the volume collapse at transition pressure is much smaller (~1.3%), which also supports the more compressible high-pressure unit cell. In

Table 2. Calculated structural parameters along with the atomic positions of NdTaO₄ and SmTaO₄ at ambient pressure and transition pressure.

NdTaO4: LP Monoclinic Phase (P2/c) @ Ambient Pressure a = 5.2504 Å; b = 5.6312 Å; c = 5.4446 Å; β = 96.79°
Nd (2e)	0.0000	0.23488	0.2500
Ta (2f)	0.5000	0.31247	0.7500
O1 (4g)	0.74508	0.90645	0.39125
O2 (4g)	0.27392	0.56319	0.49246
NdTaO4: HP Tetragonal Phase (P4/nbm) @ 40 GPa a = 5.0765 Å; c = 5.0616 Å
Nd (2b)	0.75000	0.75000	0.500
Ta (2c)	0.75000	0.25	0.0
O (8m)	0.45494	0.54506	0.23076
SmTaO4: LP Monoclinic Phase (P2/c) @ Ambient Pressure a = 5.2025 Å; b = 5.5722 Å; c = 5.3985 Å; β = 96.74°
Sm (2e)	0.0000	0.23438	0.2500
Ta (2f)	0.5000	0.30982	0.7500
O1 (4g)	0.74726	0.91026	0.39507
O2 (4g)	0.27135	0.56359	0.49281
SmTaO4: HP Tetragonal Phase (P4/nbm) @ 33 GPa a = 5.0949 Å; c = 5.0262 Å
Sm (2b)	0.75000	0.75000	0.500
Ta (2c)	0.75000	0.25	0.0
O (8m)	0.45372	0.54628	0.23465

, we have tabulated the atom positions along with their unit cell lattice parameters at ambient pressure and phase transition pressure for both the compounds. It can be clearly seen that all the constituent atoms show significant atomic rearrangement in position when compared to their ambient pressure. A more packed high-pressure unit cell is a consequence of the more effective packing of oxygen anions surrounding the Ta cation. To analyze pressure effects from this perspective, we calculated the pressure evolution of R-O and Ta-O bond distances for both the LP and HP phases. Results are summarized in Figure 7a,b, which shows calculated bond distances plotted against pressure. Figure 7a,b clearly shows that for both the compounds, at LP phase, the two largest bond distances between Nd-O decrease at a faster rate than the other six Nd-O bond lengths, whereas the Ta-O bond lengths show lower compressibility than the Nd-O bond lengths. In the high-pressure tetragonal phase, NdO₈/SmO₈ polyhedra become fully symmetric and the largest Ta-O bond shows most compressibility among all the constituent bonds, indicating the contribution of TaO₈ towards the compressibility of the unit cell.

3.3. Vibrational Properties under Compression

The primitive unit cell of the M’ fergusonite structure has two formula units, giving rise to a total of 36 phonon modes for both the compounds belonging to the P2/c space group with 2/m point group symmetry. Out of a total of 36 phonon modes, 18 modes are Raman active (8A_g + 10B_g), 15 modes are IR active (7A_u + 8B_u) and 3 are low-frequency acoustic modes. The A and B modes are one-dimensional irreducible representations which are symmetric and antisymmetric with respect to the principle axis of symmetry. The assignment of Raman and IR modes is performed in accordance with DFPT as implemented in Quantum ESPRESSO. Ambient pressure Raman and IR modes have been tabulated in

Table 3. Calculated Raman modes for NdTaO₄ and SmTaO₄ at ambient pressure along with their pressure coefficients (present work). Reported data of earlier studies on EuTaO₄ and GdTaO₄ have been included for comparison. Frequencies marked with an asterisk (*) correspond to internal modes of TaO₆ octahedra.

Raman Frequency	NdTaO4		SmTaO4		EuTaO4 [16]		GdTaO4 [17]
	ω	dω/dP	ω	dω/dP	ω	dω/dP	ω	dω/dP
Bg	99.3	1.42	101.6	1.83	102.3	1.25	100.6	1.64
Ag	108.7	1.40	107.5	2.45	107.1	1.14	103.5	1.84
Bg	119.6	1.28	119.7	1.53	119.8	1.10	119.1	1.27
Bg	138.7	2.22	138.4	2.95	138.5	1.67	138.5	2.49
Bg	164.3	0.88	168.7	1.43	172.0	0.94	171	0.92
Ag	176.7	0.76	177.2	1.05	177.6	0.54	177.9	0.41
Ag	220.6	4.07	219	4.7	216.9	4.17	211.1	4.74
Bg	257.1	2.41	261.9	2.99	263.6	2.65	261.6	2.66
Ag	257.6	2.85	262.5	3.04	264.3	2.70	263.4	2.18
Bg	314.1	4.22	321.9	5.19	325.2	4.62	322	4.78
Bg	383.5	4.20	392.5	5.3	396.0	3.96	395.3	4.80
Ag*	392.6	1.36	397.1	1.8	400.3	1.68	402.6	1.76
Bg	466.9	1.7	475.1	2.3	480.8	1.88	486.7	1.53
Ag*	472.9	1.42	478.7	1.97	482.2	1.84	487.6	1.61
Bg*	600.3	3.65	613	4.11	620.5	3.93	626.3	4.29
Ag*	609.3	3.62	620.8	4.05	627.6	3.77	633.4	4.15
Bg*	627.4	4.78	643.4	4.93	651.1	4.17	661.7	4.76
Ag*	762.2	3.55	771.5	3.58	777.1	3.11	785.9	3.69

and

Table 4. Calculated IR modes for NdTaO₄ and SmTaO₄ at ambient pressure along with their pressure coefficients (present work).

IR Frequency	NdTaO4 @ Ambient Pressure		SmTaO4 @ Ambient Pressure
	ω	dω/dP	ω	dω/dP
Au	135.5	2.13	134.3	2.88
Bu	138.6	1.58	137	2.92
Bu	156.4	3.85	155.1	5.03
Bu	207.5	−1.44	204.2	−1.54
Au	249.1	Nonlinear	253.7	Nonlinear
Bu	257.2	2.24	257.2	2.97
Bu	275.8	2.24	281.9	2.39
Au	311.1	0.87	310.8	1.53
Au	354.7	3.33	359.4	3.81
Bu	382	3.26	387.3	3.59
Bu	480.7	4.12	491.5	4.65
Au	503.6	3.74	512.4	4.23
Au	561.3	4.33	572.8	4.68
Bu	607.5	4.23	616.7	4.66
Au	743.2	3.77	752.2	3.96

, respectively. We have also included previously published Raman mode frequencies from theoretical calculations for the EuTaO₄ and GdTaO₄ compounds [16,17]. The frequency distribution of the Raman modes is quite similar to the Raman mode distribution in the wolframite structure, which belongs to the same space group P2/c as M’ fergusonite [39,40]. A total of twelve low-frequency Raman modes are present in the frequency region 100–400 cm⁻¹, two Raman modes are in the 400–600 cm⁻¹ range and four are in the higher-frequency side, 600–800 cm⁻¹. It is interesting to note that we have not observed any frequency gap as observed in the Raman spectrum of the zircon or scheelite structure [41]. This can be understood by group–subgroup relationships among zircon–scheelite–fergusonite structures by virtue of the reduction in point group symmetry from 4/mmm to 4/m to 2/m, which in turn increases the allowed numbers of Raman modes and hence fills the frequency gap. All the calculated Raman modes can be categorized as internal modes or external modes. Internal modes, lying in the higher frequency region correspond to TaO₆ octahedra while, modes at lower frequency are external modes describing the movement of rigid TaO₆ unit against the lanthanide cation [42]. Out of a total of six internal modes of TaO₆ octahedra, the 2 A_g and 2 B_g modes lie in the higher–frequency region that is from 600 to 800 cm⁻¹, and two with A_g symmetry appear in the 390–500 cm⁻¹ range [42]. Identified internal modes are marked by an asterisk in Table 3. The highest-frequency A_g mode, which appears at approximately 762 cm⁻¹ for NdTaO₄ and 771 cm⁻¹ for SmTaO₄, describes the symmetric stretching mode of TaO₆ octahedra. The frequency of the majority of the Raman active modes are observed to increase with a decrease in the lanthanum ionic radii except the two A_g modes (108 cm⁻¹ for NdTaO₄ and 107cm⁻¹ for SmTaO₄; 220 cm⁻¹ for NdTaO₄ and 219 cm⁻¹ for SmTaO₄) whereas the frequency of the two B_g modes (at 119 and 138 cm⁻¹) remains unaltered by the change in lanthanide cationic radii. This observation is valid when we extend our comparison with the published Raman mode frequencies of EuTaO₄ and GdTaO₄. Table 3 lists the pressure evolution of all the Raman active modes in P2/c structure obtained by quadratic fitting of data points under pressure. No Raman mode softening has been seen in both the compounds due to the absence of negative pressure coefficients. Hardening of all the Raman active modes has been associated with the monoclinic fergusonite phase of the other compound such as rare earth niobates GdNbO₄ and EuNbO_4, as well as rare earth tantalates GdTaO₄ and EuTaO₄ [16,17,22,23]. For rare earth vanadate, mode softening has been observed in the zircon or scheelite phase, but no mode softening has been reported in the fergusonite structure consistent with our current observation in the present work [41,43]. There is crossover between the B_g and A_g modes located at 380–400 cm⁻¹ due to a nearly 3-fold higher pressure coefficient of the B_g mode than the A_g mode. No other mode crossover has been observed in spite of the large difference in the pressure coefficient of the Raman modes. In the HP phase, stabilized in the tetragonal structure, calculation predicts the presence of the 11 Raman active modes (5 E_g + 2A_1g + 1B_1g + 3B_2g). The pressure evolution of the Raman active modes are shown in

Table 5. Calculated Raman modes for NdTaO₄ and SmTaO₄ at transition pressure along with their pressure coefficients (present work).

Raman Frequency	NdTaO4(HP) @ 40 GPa		SmTaO4(HP) @ 33 GPa
	ω	dω/dP	ω	dω/dP
Eg	103.1	Nonlinear	99.6	Nonlinear
B2g	163.6	Nonlinear	159.4	Nonlinear
Eg	208.7	1.1	199.9	1.2
B1g	357.5	0.92	343.9	1.26
Eg	412.3	1.93	397.2	2.16
A1g	412.9	Nonlinear	408.7	Nonlinear
B2g	513	2.77	501	2.73
Eg	546.6	2.9	533.6	3.09
Eg	741.8	1.55	738.4	1.85
B2g	750	1.85	743.8	2.09
A1g	823.7	1.12	815.7	1.25

. In the HP phase, all modes show positive pressure coefficients except three modes, which show nonlinear behavior under compression.

As seen in Table 4 and Table 5 out of 15 IR modes show a decreasing trend when we go from the NdTaO₄ to SmTaO₄ compound, whereas one B_u mode (257 cm⁻¹) remains unaltered. The remaining nine modes show an increase in frequency with lower lanthanide radii. From Table 4, it can be clearly seen that one IR active mode (A_u) shows a nonlinear response in frequency under compression. The presence of the negative pressure coefficient of the IR active mode (B_u) indicates phonon softening in the compound with pressure. Phonon softening was cited as among the crucial trigger points for pressure-induced instability in metal oxides [39]. The other 13 IR modes have a positive pressure coefficient. According to our calculation, the high-pressure phase for both the compounds have seven IR active modes (4E_u + 3A_2u). IR active modes at transition pressure, along with pressure coefficients, are summarized in

Table 6. Calculated IR modes for NdTaO₄ and SmTaO₄ at 40 and 33 GPa along with their pressure coefficients (present work).

IR Frequency	NdTaO4(HP)40 GPa		SmTaO4(HP)33 GPa
	ω	dω/dP	ω	dω/dP
A2u	109	Nonlinear	110.6	Nonlinear
Eu	136.2	1.92	126.2	2.24
Eu	183.2	1.36	183.8	1.33
A2u	281.2	1.80	264.1	1.89
Eu	509.3	2.26	494.6	2.52
A2u	669.1	0.62	659.6	0.85
Eu	699.9	1.53	687.7	1.72

. It can be clearly seen in Table 6 that all the modes show a positive pressure coefficient except the lowest-frequency A_2u mode, which shows a nonlinear response under pressure.

4. Conclusions

To conclude, the compressional behavior of the M’ fergusonite-structured NdTaO₄ and SmTaO₄, investigated through DFT-based first-principle simulations indicate pressure-induced first-order phase transition from monoclinic to tetragonal structure. The transition is accompanied by an increase in oxygen coordination around the Ta cation from six to eight and a nearly 1.3% volume reduction at transition pressure (40 GPa for NdTaO₄ and 33 GPa for SmTaO₄). In the low-pressure monoclinic phase, the compressibility of the unit cell has a major contribution from rare earth polyhedra, whereas both the rare earth polyhedra and tantalum polyhedra exhibit a similar contribution towards the compressibility of the unit cell for both the compounds at the HP tetragonal phase, which in turn explains the lower bulk modulus obtained in the HP phase. The pressure evolution of phonon modes has been evaluated in both the LP phase and the HP phase. No Raman mode softening has been seen in both the compounds, although one particular IR mode has been observed to show red shift under pressure, possibly leading to instability in the compounds. Earlier reported experimental high-pressure studies on EuTaO₄ and GdTaO₄ show isostructural first-order reversible phase transition at approximately 20 GPa, although theoretical calculations predict an equally probable orthorhombic Pcna as an alternative description of the HP phase near 43 and 40 GPa, respectively. The experimental and theoretical phase transition pressures differ due to the crucial dependence of a non-hydrostatic stress environment on phase transition. Therefore, a high-pressure study on NdTaO₄ and SmTaO₄ by experimental techniques is desirable for a better understanding of structural and vibrational changes under compression, which is out of scope in the present work.