# Substitutions in Nd5−xTbxMo3O16+δ Series: Phase Formation, Atomistic Modeling, and Crystal Structure of Nd5Mo3O16+δ-Based Solid Solutions

Authors: Konstantin A. Chebyshev; Nelly I. Selikova; Lyudmila V. Pasechnik; A. V. Ignatov
Journal: Journal of Chemistry
 (2023)

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## Abstract

Samples of the Nd5−xTbxMo3O16+δ series were obtained by solid-state synthesis from metal oxides at 1050°C. The formation of solid solutions based on cubic and monoclinic phases and a two-phase region between them was observed in the Nd5−xTbxMo3O16+δ compositions. Increasing the terbium content in the system leads to a decrease in the unit cell parameters of the cubic and monoclinic phases within their homogeneity regions, which confirms the formation of solid solutions. It has been established that the terbium oxidation state in the Nd5Mo3O16+δ crystal structure is +3. The predominant placement of terbium atoms in the 8c position is observed by crystal structure refinement and confirmed by the results of the atomistic simulation. The introduction of terbium into the crystal structure of neodymium molybdate leads to a decrease in the Ln1–O1 and Ln2–O2 interatomic distances. The atomistic simulation was performed by the GULP program using the fit potential of the terbium ion. Terbium molybdate with Tb5Mo3O16+δ composition is a subtraction solid solution based on Tb2MoO6. Increasing the unit cell parameters of Tb5Mo3O16+δ monoclinic phase compared to Tb2MoO6 was confirmed by structure refinement and atomistic simulation.


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## Body

## 1. Introduction

Fluorite-like molybdates of rare earth elements (REE) with the Ln5Mo3O16+δ composition were studied in recent years due to their high conductivity values at medium temperatures [1–3]. These compounds have an ionic or electron-ionic type of conductivity [1, 4], depending on the nature of REE and external conditions (temperature and partial pressure of oxygen). Among these compounds, neodymium and praseodymium molybdates are the most stable [5, 6]. For example, Pr5Mo3O16+δ is a promising electrode material for medium-temperature solid oxide fuel cells [7, 8] due to the presence of an electronic component of conductivity probably due to the existence of praseodymium in various oxidation states (+3 and +4). Neodymium molybdate is an ionic conductor that demonstrates a noticeable electronic conductivity at high temperatures [2, 4, 9].The crystal structure of Ln5Mo3O16+δ fluorite-like molybdates is derived from the fluorite structure and described in the cubic system with the space group Pn − 3n, Z = 4 [10]. The unit cell is a double fluorite one (a = 2af), which is confirmed by the observation of superstructure reflections in XRD patterns. The formation of the superstructure is explained by the arrangement of molybdenum and REE atoms over the cationic positions of the fluorite structure. REE atoms occupy two crystallographically unequal positions, 12e (x, 0.25, 0.25), and 8c (0, 0, 0), while molybdenum is placed in position 12 d (0, 0.75, 0.25). The oxygen environment of REE is distorted cubes, and molybdenum forms almost regular tetrahedra.Theδ value in the formula depends on the synthesis conditions of REE molybdates [1]. In the compounds obtained under reducing conditions,, one-third of the molybdenum atoms have a +5 oxidation state, and the oxygen content in the formula is 16 (value δ = 0). The oxidizing of “reduced” compositions or obtaining them in the air leads to an increase in the δ value up to 0.5 (total oxygen content is 16.5). In the Ln5Mo3O16+δ structure, the number of anionic positions is related to the number of cationic ones as 2 to 1; however, at δ = 0.5, the ratio of the number of anions to the sum of REE and molybdenum atoms is 2.0625. Thus, there are two atoms of “overstoichiometric” oxygen in relation to the number of crystallographic positions per unit cell. The authors of [1] suggested that “overstoichiometric” oxygen δ is located in the octahedral voids of the fluorite structure, causing high ionic conductivity of these compounds. In addition, the results of the refinement of the crystal structure indicate the possibility of incomplete filling of the 48i [9] position, and in this case, the value of δ can be negative.The study of substitutions in the structure of neodymium molybdate is associated with the possibility of increasing the ionic or electronic component of conductivity. Substitution of samarium for neodymium leads to an increase in conductivity, which is highest for the composition with the atomic ratio Nd : Sm ≈ 1 : 1 [11]. A similar dependence was observed for lead-containing solid solutions with the Ln5Mo3O16+δ structure [12]. Substitution of neodymium or molybdenum for elements with a lower charge does not lead to an increase in conductivity [3, 13, 14]. Previously, we studied the substitution of neodymium for a number of elements, including lanthanum, cerium, praseodymium, samarium, erbium, bismuth, and lead [11, 13, 15–17]. It was shown that the most promising way to enhance the conductivity is to replace neodymium with large cations (lanthanum, praseodymium) or cerium and thorium [15, 18], which can exhibit a +4 oxidation state in oxide compounds. Among rare earth elements, terbium can also exhibit oxidation states of +4 [19], so its introduction into the crystal structure of neodymium molybdate can lead to a rise in conductivity. In addition, terbium-contained compounds exhibit catalytic properties [20, 21]. This paper presents a study of the isomorphic substitution of neodymium with terbium in the Nd5Mo3O16 +δ structure.


## 2. Materials and Methods

### 2.1. Synthesis

Samples of the Nd5−xTbxMo3O16+δ series (x = 0; 0.05; 0.1; 0.15; 0.3; 0.5; 0.7; 1.0; 2.0; 3.0; 4.0; 5.0) were obtained by conventional solid-state synthesis using Nd2O3, Tb4O7, and MoO3 oxides (purity ≥ 99.9%). REE and molybdenum oxides were calcined, respectively, at 1000°С and 500°С before homogenization by using a small quantity of ethanol. Samples were annealed at 500°С for fixing MoO3 and at 800°С, followed by homogenization in an agate mortar.Dense polycrystalline samples for conductivity measurements were obtained from powders calcined at 800°C, which were pressed with the addition of 5% water solution of polyvinyl alcohol into 8 mm diameter disks. The sintering temperature was 1050°C for 20 h. The relative density of ceramics was about 90%. Electrodes were applied to the surface of ceramic pellets by burning silver-containing paste.


### 2.2. X-Ray Diffraction

XRD analysis was performed using a DRON-2 diffractometer, CuKα-radiation, and a Ni filter. To refine the crystal structure by the Rietveld method we used data collected on a Rigaku Ultima IV diffractometer using filtered CuKα radiation (Ni filter) with a step of 0.02° 2Θ and a scanning rate of 0.7° 2Θ/min in the range 10°–140° 2Θ. Refinement was performed using the FULLPROF.2k program (version 5.30) with the WinPLOTR graphical interface [22]. The unit cell parameters were calculated by full-profile analysis of XRD patterns by the Le Bail method. The data on the crystal structure of Nd5Mo3O16 neodymium molybdate were used as the initial model for refinement [10].Determination of element content was performed by scanning electron microscopy on a JSM-6490LV X-ray microscope using an INCA Penta FETx3 energy-dispersive spectrometer.


### 2.3. Atomistic Simulation

Atomistic modeling was carried out using the GULP 4.0 (General Utility Lattice Program) program by minimizing the energy of interatomic interactions [23]. Interatomic potentials Uij were set by the sum of the Coulomb interaction and the Buckingham potential (buck), which takes into account the repulsion of atoms according to Born–Mayer and dispersion interactions(1)Uij=qiqje2Rij+Aijexp−Rijρij−cijRij6.The polarization of oxygen anions was described using the “shell” model, where the entire mass of the ion is attributed to a positively charged core surrounded by a “shell.” The core and the shell are connected by elastic interactions (spring):(2)Uijs=12χili2.The region of action of two-particle potentials was limited by a distance of 15 Å. The parameters of potential values were taken from [24] and refined according to the crystal structure data for neodymium molybdate Nd5Mo3O16+δ. The initial potential for the interatomic interaction Tbc–Os [25] was refined using the crystal structure of terbium molybdate, Tb2MoO6, determined from neutron diffraction data [26]. The final values of the parameters of the potentials and charges of the particles are given in Table 1.Table 1
Parameters of pair interaction potential and charges of atomic cores (c) and shells (s).
InteractionPotentialParameters of potentialAtomChargeA (eV)λ (Å)c (eV Å)Ndc–OsBuck5617.220.2682640.0Ndc2.55Moc–OsBuck945.9470.3666170.0Moc5.1Tbc–OsBuck3289.6790.28870.0Tb2.55Os–OsBuck508.2226990.31483826.8965Oc0.7465Oc–OsSpringχ = 52.11414 eV/Å2Os−2.4997The defect energy was calculated by the Mott–Littleton method [27], with the defect sphere radius r1 = 10 Å and the screening layer thickness r1 − r2 being equal to 20 Å.


### 2.4. Electrical Conductivity

The electrical conductivity was determined using an LCR DE-5000 meter at a frequency of 1 kHz in the temperature range of 300–700°C with a step of 10°C at a heating rate of 2°C/min. The activation energy was calculated from the linear region of the obtained dependency.


## 2.1. Synthesis

Samples of the Nd5−xTbxMo3O16+δ series (x = 0; 0.05; 0.1; 0.15; 0.3; 0.5; 0.7; 1.0; 2.0; 3.0; 4.0; 5.0) were obtained by conventional solid-state synthesis using Nd2O3, Tb4O7, and MoO3 oxides (purity ≥ 99.9%). REE and molybdenum oxides were calcined, respectively, at 1000°С and 500°С before homogenization by using a small quantity of ethanol. Samples were annealed at 500°С for fixing MoO3 and at 800°С, followed by homogenization in an agate mortar.Dense polycrystalline samples for conductivity measurements were obtained from powders calcined at 800°C, which were pressed with the addition of 5% water solution of polyvinyl alcohol into 8 mm diameter disks. The sintering temperature was 1050°C for 20 h. The relative density of ceramics was about 90%. Electrodes were applied to the surface of ceramic pellets by burning silver-containing paste.


## 2.2. X-Ray Diffraction

XRD analysis was performed using a DRON-2 diffractometer, CuKα-radiation, and a Ni filter. To refine the crystal structure by the Rietveld method we used data collected on a Rigaku Ultima IV diffractometer using filtered CuKα radiation (Ni filter) with a step of 0.02° 2Θ and a scanning rate of 0.7° 2Θ/min in the range 10°–140° 2Θ. Refinement was performed using the FULLPROF.2k program (version 5.30) with the WinPLOTR graphical interface [22]. The unit cell parameters were calculated by full-profile analysis of XRD patterns by the Le Bail method. The data on the crystal structure of Nd5Mo3O16 neodymium molybdate were used as the initial model for refinement [10].Determination of element content was performed by scanning electron microscopy on a JSM-6490LV X-ray microscope using an INCA Penta FETx3 energy-dispersive spectrometer.


## 2.3. Atomistic Simulation

Atomistic modeling was carried out using the GULP 4.0 (General Utility Lattice Program) program by minimizing the energy of interatomic interactions [23]. Interatomic potentials Uij were set by the sum of the Coulomb interaction and the Buckingham potential (buck), which takes into account the repulsion of atoms according to Born–Mayer and dispersion interactions(1)Uij=qiqje2Rij+Aijexp−Rijρij−cijRij6.The polarization of oxygen anions was described using the “shell” model, where the entire mass of the ion is attributed to a positively charged core surrounded by a “shell.” The core and the shell are connected by elastic interactions (spring):(2)Uijs=12χili2.The region of action of two-particle potentials was limited by a distance of 15 Å. The parameters of potential values were taken from [24] and refined according to the crystal structure data for neodymium molybdate Nd5Mo3O16+δ. The initial potential for the interatomic interaction Tbc–Os [25] was refined using the crystal structure of terbium molybdate, Tb2MoO6, determined from neutron diffraction data [26]. The final values of the parameters of the potentials and charges of the particles are given in Table 1.Table 1
Parameters of pair interaction potential and charges of atomic cores (c) and shells (s).
InteractionPotentialParameters of potentialAtomChargeA (eV)λ (Å)c (eV Å)Ndc–OsBuck5617.220.2682640.0Ndc2.55Moc–OsBuck945.9470.3666170.0Moc5.1Tbc–OsBuck3289.6790.28870.0Tb2.55Os–OsBuck508.2226990.31483826.8965Oc0.7465Oc–OsSpringχ = 52.11414 eV/Å2Os−2.4997The defect energy was calculated by the Mott–Littleton method [27], with the defect sphere radius r1 = 10 Å and the screening layer thickness r1 − r2 being equal to 20 Å.


## 2.4. Electrical Conductivity

The electrical conductivity was determined using an LCR DE-5000 meter at a frequency of 1 kHz in the temperature range of 300–700°C with a step of 10°C at a heating rate of 2°C/min. The activation energy was calculated from the linear region of the obtained dependency.


## 3. Results and Discussion

### 3.1. X-Ray Diffraction

Figure1 shows XRD patterns of polycrystalline samples of the Nd5−xTbxMo3O16+δ series. According to the results of phase analysis formation of solid solutions based on the cubic phase with the Nd5Mo3O16+δ structure in the range of terbium content x ˂ 2 is observed, however, single-phase samples were obtained for compositions x < 0.7. Weak reflections of the impurity phase appear on the X-ray diffraction pattern of the x = 0.7 sample. The composition range of 0.7 ≤ x ≤ 2 relates to the two-phase area.Figure 1
X-ray powder diffraction patterns of the Nd5−xTbxMo3O16+δ polycrystalline samples. (a) Nd5Mo3O16+δ reflections. (b) Tb2MoO6 reflections.The compositionx = 5 falls obviously within the region of homogeneity of monoclinic Tb2MoO6 terbium molybdate because of the XRD pattern described in the monoclinic system with the С2/с space group (Rp = 7.06, Rwp = 9.15, Rexp = 7.69, RB = 13.3, Rf = 19.6, χ2 = 1.42) and is similar to Tb2MoO6 unit cell parameters (Figure 2). The region of existence of solid solutions of cubic neodymium molybdate in terbium molybdate with a monoclinic structure is 0.7 ≤ x ≤ 5, whereas the concentration region where all observed reflections are described in the monoclinic structure is 2 ≤ x ≤ 5.Figure 2
Experimental (red dots) and calculated (black profile) XRD patterns, their difference (blue profile), and Bragg reflection positions (green lines) of Tb5Mo3O16+δ.It is known that terbium in oxide compounds can exhibit oxidation states of 3+ and 4+ [19]. Depending on the oxidation state of the ion included in the structure, the substitution process will be different (described using Kröger–Vink notation)(3)NdNd×+14Tb4O7⟶TbNd×+12Nd2O3+18O2↑,(4)NdNd×+14Tb4O7+18O2⟶TbNd•+12Nd2O3+12Oi″↑.One can expect a decrease in the unit cell parameter in both cases of terbium oxidation state. According to scheme (1), it occurs due to the smaller size of the Tb3+ ion entering the structure instead of Nd3+ (for c.n. = 8 ri (Nd3+) = 1.109 Å; ri (Tb3+) = 1.040 Å) [28]. If the crystal structure of neodymium molybdate Nd5Mo3O16+δ includes terbium with an oxidation state of 4+ (scheme 2), we can also expect a decrease in the cell parameter, as was observed in previously studied systems with cerium Ln5−xCexMo3O16+δ (Ln = La [18], Nd [15]).Isomorphous substitution of neodymium for terbium leads to a decrease in the unit cell parameter of solid solutions with a cubic structure in the concentration rangex ≤ 0.5 (Figure 3). In the two-phase region, the lattice parameter does not change within the error of determination, which indicates the completion of the replacement of neodymium by terbium in the crystal structure of Nd5Mo3O16+δ. The substitution limit in the Nd5−xTbxMo3O16+δ system is determined by the inflection of the dependence of the unit cell parameter on the composition, which is x ≈ 0.5. The value of limit substitution was refined by extrapolation of (−421) reflection intensity dependence on the series content is x = 0.52.Figure 3
Cubic phase unit cell parameter variation with the Nd5−xTbxMo3O16+δ series content (insert: dependence of (−421) reflection intensity on the series content).The formation of solid solutions based on terbium molybdate with monoclinic structure was confirmed by a linear increase in unit cell parameters with a rise in the neodymium content (Figure4). In the two-phase region, the cell parameters cease to change, which is associated with the completion of the replacement of terbium by neodymium. The limit of substitution of terbium by neodymium in the monoclinic phase, determined from the inflection of the dependence of the unit cell volume, is х ≈ 1.2.Figure 4
Unit cell parameter and cell volume variation of monoclinic phase with the Nd5−xTbxMo3O16+δ series content.Three ranges were observed on the dependence of the volume per one formula unit of fluorite (MeO2) in the Nd5−xTbxMo3O16+δ system (Figure 5) that correspond to the regions of homogeneity of the cubic (c) and monoclinic (m) phases, as well as the region of their mixtures (c + m). The monoclinic phase is denser than the cubic; accordingly, it can be expected an increase in stability of the monoclinic structure with pressure increasing.Figure 5
Fluorite formula’s volume variation with the Nd5−xTbxMo3O16+δ series content.The substitution limit should decrease with increasing differences in the size of the substituting units, according to the theory of isomorphic miscibility. A certain substitution limit in the series under study is greater than that for the same cerium-doped solid solutions [15, 18] and slightly less than that for the Nd5−xDyxMo3O16+δ series [29] (Table 2), which indirectly indicates that terbium enters the crystal structure of neodymium molybdate mainly in the 3+ oxidation state.Table 2
Substitution limits of the Nd5−xLnxMo3O16+δ series, where Ln = Ce, Dy, Tb, and ionic radii of doping elements.
Ce4+Dy3+Tb3+Tb4+ri (Å)0.971.0271.040.88Subst. limit (x)0.330.540.52Isomorphic substitution of neodymium by terbium can be described using the Kröger–Vink notation as follows:(5)NdNd×+14Tb4O7⟶TbNd×+12Nd2O3+18O2↑,where we used the most stable terbium oxide at standard conditions as a doping ion source. In this case, terbium undergoes a transition from the oxidation state 4+ in the mixed oxide Tb4O7 to 3+ in the solid solution. Then, the replacement of neodymium by terbium can be represented by the transition of the oxide Tb4O7 to Tb2O3 and subsequent substitution(6)12Tb4O7⟶Tb2O3+14O2↑;∆HI,(7)NdNd×+12Tb2O3⟶TbNd×+12Nd2O3;∆HII,and the enthalpy of substitution is the sum of the enthalpies of these processes. Accordingly, the enthalpy of substitution of neodymium for terbium is greater by the value of the enthalpy of reduction of Tb4O7 oxide to Tb2O3 (equation (6)) compared to rare earth elements having stable oxides of composition Ln2O3. The calculation according to [30] shows that the reduction process of terbium oxide Tb4O7 is endothermic (ΔHf(Tb2O3) = −443.172 kcal/mol (−9.61 eV per terbium atom); ΔHf(Tb4O7) = −919.2 kcal/mol (−9.95 eV); ΔHI = −9.61—(−9.95) = 0.34 eV), which is the possible reason for the slightly lower substitution limit for neodymium with terbium compared to dysprosium.


### 3.2. SEM

For the Nd4Tb1Mo3O16+δ sample belonging to the two-phase region, after annealing at 1050°C, small crystallites with a size of less than 1 μm and larger ones reaching 100 μm were observed according to SEM data (Figure 6). Small particles in the image have a lighter shade than large ones due to the high content of heavy elements.Figure 6
Microstructure image of the Nd4Tb1Mo3O16+δ sample.The content of chemical elements in the sample for particles of different sizes according to energy-dispersive X-ray spectroscopy data is given in Table3. It should be noted that the experimentally determined composition (from the scanning of the full image surface) of the sample is in good agreement with the calculated values from the formula. For spectra 1 and 2, which correspond to small particles, the ratio of the content of the total REE to molybdenum is ≈1.96, and for spectra 3, 4 (large particles) ≈1.54. With respect to the calculated composition, for which n(REE): n(Mo) ≈1.67, small particles are enriched in rare earth elements, and their composition is close to the formula Ln2MoO6. Thus, small particles can be assigned to a phase with a monoclinic structure, while large ones to a cubic fluorite-like one.Table 3
Chemical element content of the Nd4Tb1Mo3O16+δ sample, at %.
ElementCalcExp. (total)Exp. (small particles, spectra 1, 2)Exp. (large particles, spectra 3, 4)Nd16.316.920.017.2Tb4.14.07.23.4Mo12.212.713.913.4O67.366.458.966.0


### 3.3. Terbium Oxidation State

The value of the substitution limit indicates that terbium exists in the crystal structure of neodymium molybdate predominantly in oxidation state 3+. The chemical determination of the REE content in higher oxidation states is usually carried out using iodometric or permanganometric titration. To confirm the degree of oxidation of terbium in the studied fluorite-like molybdates, a weighed portion of the Nd4.5Tb0.5Mo3O16+δ solid solution was dissolved in sulfuric acid with an excess of manganese sulfate (II), as described in [31]. Under these conditions, Tb4+ ions oxidize manganese (II) to a permanganate ion, which causes a violet color in the solution(8)5Tb4++Mn2++4H2O⟶MnO4−+5Tb3++8H+.We did not observe the appearance of solution color upon dissolution of terbium-containing molybdates; that would prove the existence of terbium in solid solutions only in the 3+ oxidation state.


### 3.4. Rietveld Refinement

The crystal structure refinement of the Nd4.85Tb0.15Mo3O16+δ solid solution was carried out by the Rietveld method using X-ray diffraction data (Figure 7).Figure 7
Experimental (red dots) and calculated (black profile) XRD patterns, their difference (blue profile), and Bragg reflection positions (green lines) of the Nd4.85Tb0.15Mo3O16+δ solid solution.As a result of the calculation (Table4), the predominant location of the substituting element in the Ln2 position was obtained, as in the case of systems with rare earth elements smaller than neodymium [17]. However, the error in the fitted value of site occupancies is much larger than the actual content of terbium in the sample; therefore, such an occupation of lanthanide sites was fixed and was not further refined. The isotropic thermal parameters were refined separately for each position of the cations and in total for all oxygen atoms. Similar to the previous studies [6, 9–11], the value of Biso for atoms in position Ln2 is larger than that of position Ln1. The reliability factors for the last stage of refinement were: RB = 3.03; Rf = 4.68; Rp = 9.35; Rwp = 13.8; Rexp = 10.26; χ2 = 1.81.Table 4
Refined coordinates and isotropic thermal parameters of atoms, as well as the occupancy of positions of Nd4.85Tb0.15Mo3O16+δ solid solution.
АtomxyZBisoOcc.PositionNd10.00939 (2)0.250.250.22 (5)1212eTb10.00939 (2)0.250.250.22 (5)0Nd20001.22 (9)7.48cTb20001.22 (9)0.6Mo00.750.250.53 (7)1212dO10.1213 (11)0.1213 (11)0.1213 (11)1.02 (19)1616fO20.5827 (11)0.3638 (8)0.8291 (10)1.02 (19)4848iThe interatomic distances and bond valence sums (BVS) were calculated based on the refined atomic coordinates (Table5). The introduction of terbium into the neodymium molybdate structure leads to a decrease in the Ln1–O1 and Ln2–O2 distances, while the Ln1–O2 and Ln2–O1 distances slightly increase, which indicates a distortion of the crystal structure upon modification with a smaller ion than neodymium.Table 5
Some interatomic distances and bond valence sums (BVS) for Nd4.85Tb0.15Mo3O16+δ (∗) solid solution compared to Nd5Mo3O16+δ (∗∗).
PositionEnvironmentBVSr (Me– O) (Å)∗r (Me– O) (Å)∗∗Nd14 × O13.09 (4)2.356 (16)2.377 (3)Tb14 × O22.53 (4)2.597 (11)2.567 (3)Nd22 × O12.79 (3)2.320 (15)2.283 (2)Tb26 × O22.29 (3)2.578 (14)2.615 (2)Mo4 × O25.62 (11)1.781 (13)1.768 (2)The suitability of crystallographic positions for accommodating various ions and the stability of crystal structure can be assessed using the bond valence method [32].The BVS of terbium ions is significantly lower than the expected value of +3 in both positions. This result suggests that Tb–O bonds are under tensile stress, giving rise to the instability of the crystal structure. At the same time, the deviation of BVS for cations from the oxidation state for the Ln1 position is smaller than for the Ln2 one, which indicates that the Ln1 position is more suitable for the location of both terbium and neodymium ions. Nevertheless, according to the refinement of the occupancy of the crystallographic positions, the terbium atoms are located in the Ln2 polyhedron. This is probably due to the strong distortion of the Ln2 polyhedron, which is characterized by the shortest Ln2–O1 distances, making it difficult to locate a large Nd3+ ion in this position.


### 3.5. Conducting Properties

Figure8 shows the temperature dependence of the specific conductivity of neodymium molybdate and the Nd4.7Tb0.3Mo3O16+δ solid solution. Low-temperature parts of the dependence of terbium-contained sample can be attributed to impurity conductivity and high-temperature area to intrinsic one.Figure 8
Temperature dependence of conductivity for samples of the Nd5Mo3O16+δ molybdate () and the Nd4.7Tb0.3Mo3O16+δ solid solution ().Doping of neodymium molybdate by terbium leads to decreasing conductivity within the entire temperature range. It is known that the arrangement of ions with a variable oxidation state in equivalent positions should lead to a significant contribution of electronic conductivity [33]. For the modified samples of the Nd5−xTbxMo3O16+δ series, a decrease in conductivity values indicates the existence of terbium only in the +3 oxidation state in solid solutions with a fluorite-like structure.


### 3.6. Atomistic Modeling

The atomic coordinates and interatomic distances calculated as a result of the Nd5Mo3O16+δ molybdate structure optimization are in satisfactory agreement with the experimental values (Table 6). The calculated unit cell parameter is close to the value for neodymium molybdate obtained under reducing conditions [1, 10, 34]. The slightly smaller values of interatomic distances obtained from the simulation results compared with the experimental data are due to the smaller unit cell parameter, which is obviously due to the absence of “overstoichiometric” oxygen in the model.Table 6
Experimental and calculated unit cell parameters and interatomic distances of Nd5Mo3O16+δ crystal structure.
CalcExp∗Exp [9]Exp [34]a (Å)10.972511.03916 (16)11.03379 (9)10.9959 (7)Interatomic distances (Å)Nd1–O1 × 42.3462.377 (3)2.365 (1)2.3612 (17)Nd1–O2 × 42.5562.567 (3)2.578 (1)2.566 (3)Nd2–O1 × 22.2342.283 (2)2.291 (2)2.293 (4)Nd2–O2 × 62.6092.6149 (18)2.608 (1)2.569 (3)Mo–O2 × 41.7711.767 (2)1.776 (1)1.791 (4)∗Present work.The usage of refined values of the Buckingham potential for the Tb3+ ion also makes it possible to satisfactorily describe the crystal structure of terbium molybdates (Table 7). Attention is drawn to the larger values of the unit cell parameters obtained from the full-profile analysis of the X-ray diffraction pattern for Tb5Mo3O16.5 with a monoclinic structure compared to Tb2MoO6 molybdate [26]. Since Tb5Mo3O16.5 molybdate ( composition can be represented as Tb5/3MoO5.5) has the same structure as Tb2MoO6, it contains less terbium than Tb2MoO6, which leads to the formation of vacancies in the terbium and oxygen positions. In this case, one would expect a decrease in the unit cell parameters with a decrease in the terbium content. However, the simulation results show that for the model containing vacancies at the sites of terbium and oxygen, the unit cell parameters are also larger than for Tb2MoO6 molybdate, which agrees satisfactorily with the experimental values. The discrepancy between the calculated parameters of the crystal structure and the experimental ones may be due to the fact that in the model, the vacancies at the sites of terbium and oxygen were statistically distributed between the positions of the crystal structure.Table 7
Experimental and calculated parameters of monoclinic terbium molybdate crystal structure.
Tb5Mo3O16.5 expTb5Mo3O16.5 calcTb2MoO6 [26]Tb2MoO6 calcUnit cell parametersa (Å)16.493 (3)16.557516.4554 (3)16.4748b (Å)11.150 (2)11.092911.1143 (2)11.0421c (Å)5.4060 (12)5.44365.3936 (1)5.3847β (°)108.442 (7)109.79108.428 (1)109.68ρ (g/cm3)6.366.347.247.34Interatomic distances (Å)Tb1–O4 × 22.77 (3)2.9682.770 (5)2.863Tb1–O5 × 22.30 (3)2.3102.319 (5)2.330Tb1–O6 × 22.22 (2)2.2422.223 (5)2.248Tb1–O6 × 22.39 (3)2.4122.387 (5)2.335Tb2–O2 × 22.42 (4)2.3362.377 (6)2.352Tb2–O2 × 22.491 (19)2.5232.488 (4)2.484Tb2–O4 × 22.516 (17)2.3902.478 (5)2.398Tb2–O5 × 22.25 (3)2.3612.238 (5)2.297Tb3–O1 × 12.57 (4)2.5202.499 (5)2.461Tb3–O1 × 12.63 (3)2.5452.570 (5)2.477Tb3–O3 × 12.31 (3)2.4512.426 (6)2.373Tb3–O3 × 12.41 (4)2.4582.435 (5)2.445Tb3–O4 × 12.62 (4)2.8502.655 (5)2.783Tb3–O5 × 12.34 (3)2.2452.265 (6)2.270Tb3–O6 × 12.21 (4)2.1832.213 (6)2.198Tb3–O6 × 12.38 (3)2.4232.273 (5)2.359Mo–O1 × 11.74 (4)1.7681.796 (5)1.765Mo–O2 × 11.74 (4)1.7791.799 (6)1.775Mo–O3 × 11.79 (3)1.8021.812 (6)1.801Mo–O4 × 11.84 (4)1.7971.805 (6)1.796Mo–O5 × 12.15 (4)1.9692.214 (6)2.085Mo–O1 × 13.02 (4)3.1353.113 (5)3.088Mo–O2 × 13.17 (4)3.0813.072 (5)3.027Mo–O3 × 13.01 (4)2.9712.983 (6)2.970The Mott-Littleton method was used to calculate the energies of substitution defects of neodymium with terbium in positions 12e and 8c. The simulation results show that terbium placement in the Ln2 (8c) position of the neodymium molybdate crystal structure is energetically more favorable (Table8).Table 8
Energy of substitutional defect (Ed (NdTb×)) and substitution process according to equation (1) (Es), eV.
PositionLn1Ln2Ed (NdTb×)0.500.43Es1.091.02The quantitative relationship between the probabilities of terbium placement in positions Ln1 (12e) and Ln2 (8c) and the calculated defect energies E1 and E2, respectively, can be determined using the Boltzmann distribution in the form(9)P2P1=eE1−E2/−kT,where P1 and P2 are the probabilities of terbium placing in crystallographic position with defect energies E1 and E2, respectively; T is the synthesis temperature (1323 K), k is the Boltzmann constant (8.617·10–5 eV/K). The resulting ratio of the terbium content in the Ln2 and Ln1 positions calculated by the substitutional defect energies is 1.84.


## 3.1. X-Ray Diffraction

Figure1 shows XRD patterns of polycrystalline samples of the Nd5−xTbxMo3O16+δ series. According to the results of phase analysis formation of solid solutions based on the cubic phase with the Nd5Mo3O16+δ structure in the range of terbium content x ˂ 2 is observed, however, single-phase samples were obtained for compositions x < 0.7. Weak reflections of the impurity phase appear on the X-ray diffraction pattern of the x = 0.7 sample. The composition range of 0.7 ≤ x ≤ 2 relates to the two-phase area.Figure 1
X-ray powder diffraction patterns of the Nd5−xTbxMo3O16+δ polycrystalline samples. (a) Nd5Mo3O16+δ reflections. (b) Tb2MoO6 reflections.The compositionx = 5 falls obviously within the region of homogeneity of monoclinic Tb2MoO6 terbium molybdate because of the XRD pattern described in the monoclinic system with the С2/с space group (Rp = 7.06, Rwp = 9.15, Rexp = 7.69, RB = 13.3, Rf = 19.6, χ2 = 1.42) and is similar to Tb2MoO6 unit cell parameters (Figure 2). The region of existence of solid solutions of cubic neodymium molybdate in terbium molybdate with a monoclinic structure is 0.7 ≤ x ≤ 5, whereas the concentration region where all observed reflections are described in the monoclinic structure is 2 ≤ x ≤ 5.Figure 2
Experimental (red dots) and calculated (black profile) XRD patterns, their difference (blue profile), and Bragg reflection positions (green lines) of Tb5Mo3O16+δ.It is known that terbium in oxide compounds can exhibit oxidation states of 3+ and 4+ [19]. Depending on the oxidation state of the ion included in the structure, the substitution process will be different (described using Kröger–Vink notation)(3)NdNd×+14Tb4O7⟶TbNd×+12Nd2O3+18O2↑,(4)NdNd×+14Tb4O7+18O2⟶TbNd•+12Nd2O3+12Oi″↑.One can expect a decrease in the unit cell parameter in both cases of terbium oxidation state. According to scheme (1), it occurs due to the smaller size of the Tb3+ ion entering the structure instead of Nd3+ (for c.n. = 8 ri (Nd3+) = 1.109 Å; ri (Tb3+) = 1.040 Å) [28]. If the crystal structure of neodymium molybdate Nd5Mo3O16+δ includes terbium with an oxidation state of 4+ (scheme 2), we can also expect a decrease in the cell parameter, as was observed in previously studied systems with cerium Ln5−xCexMo3O16+δ (Ln = La [18], Nd [15]).Isomorphous substitution of neodymium for terbium leads to a decrease in the unit cell parameter of solid solutions with a cubic structure in the concentration rangex ≤ 0.5 (Figure 3). In the two-phase region, the lattice parameter does not change within the error of determination, which indicates the completion of the replacement of neodymium by terbium in the crystal structure of Nd5Mo3O16+δ. The substitution limit in the Nd5−xTbxMo3O16+δ system is determined by the inflection of the dependence of the unit cell parameter on the composition, which is x ≈ 0.5. The value of limit substitution was refined by extrapolation of (−421) reflection intensity dependence on the series content is x = 0.52.Figure 3
Cubic phase unit cell parameter variation with the Nd5−xTbxMo3O16+δ series content (insert: dependence of (−421) reflection intensity on the series content).The formation of solid solutions based on terbium molybdate with monoclinic structure was confirmed by a linear increase in unit cell parameters with a rise in the neodymium content (Figure4). In the two-phase region, the cell parameters cease to change, which is associated with the completion of the replacement of terbium by neodymium. The limit of substitution of terbium by neodymium in the monoclinic phase, determined from the inflection of the dependence of the unit cell volume, is х ≈ 1.2.Figure 4
Unit cell parameter and cell volume variation of monoclinic phase with the Nd5−xTbxMo3O16+δ series content.Three ranges were observed on the dependence of the volume per one formula unit of fluorite (MeO2) in the Nd5−xTbxMo3O16+δ system (Figure 5) that correspond to the regions of homogeneity of the cubic (c) and monoclinic (m) phases, as well as the region of their mixtures (c + m). The monoclinic phase is denser than the cubic; accordingly, it can be expected an increase in stability of the monoclinic structure with pressure increasing.Figure 5
Fluorite formula’s volume variation with the Nd5−xTbxMo3O16+δ series content.The substitution limit should decrease with increasing differences in the size of the substituting units, according to the theory of isomorphic miscibility. A certain substitution limit in the series under study is greater than that for the same cerium-doped solid solutions [15, 18] and slightly less than that for the Nd5−xDyxMo3O16+δ series [29] (Table 2), which indirectly indicates that terbium enters the crystal structure of neodymium molybdate mainly in the 3+ oxidation state.Table 2
Substitution limits of the Nd5−xLnxMo3O16+δ series, where Ln = Ce, Dy, Tb, and ionic radii of doping elements.
Ce4+Dy3+Tb3+Tb4+ri (Å)0.971.0271.040.88Subst. limit (x)0.330.540.52Isomorphic substitution of neodymium by terbium can be described using the Kröger–Vink notation as follows:(5)NdNd×+14Tb4O7⟶TbNd×+12Nd2O3+18O2↑,where we used the most stable terbium oxide at standard conditions as a doping ion source. In this case, terbium undergoes a transition from the oxidation state 4+ in the mixed oxide Tb4O7 to 3+ in the solid solution. Then, the replacement of neodymium by terbium can be represented by the transition of the oxide Tb4O7 to Tb2O3 and subsequent substitution(6)12Tb4O7⟶Tb2O3+14O2↑;∆HI,(7)NdNd×+12Tb2O3⟶TbNd×+12Nd2O3;∆HII,and the enthalpy of substitution is the sum of the enthalpies of these processes. Accordingly, the enthalpy of substitution of neodymium for terbium is greater by the value of the enthalpy of reduction of Tb4O7 oxide to Tb2O3 (equation (6)) compared to rare earth elements having stable oxides of composition Ln2O3. The calculation according to [30] shows that the reduction process of terbium oxide Tb4O7 is endothermic (ΔHf(Tb2O3) = −443.172 kcal/mol (−9.61 eV per terbium atom); ΔHf(Tb4O7) = −919.2 kcal/mol (−9.95 eV); ΔHI = −9.61—(−9.95) = 0.34 eV), which is the possible reason for the slightly lower substitution limit for neodymium with terbium compared to dysprosium.


## 3.2. SEM

For the Nd4Tb1Mo3O16+δ sample belonging to the two-phase region, after annealing at 1050°C, small crystallites with a size of less than 1 μm and larger ones reaching 100 μm were observed according to SEM data (Figure 6). Small particles in the image have a lighter shade than large ones due to the high content of heavy elements.Figure 6
Microstructure image of the Nd4Tb1Mo3O16+δ sample.The content of chemical elements in the sample for particles of different sizes according to energy-dispersive X-ray spectroscopy data is given in Table3. It should be noted that the experimentally determined composition (from the scanning of the full image surface) of the sample is in good agreement with the calculated values from the formula. For spectra 1 and 2, which correspond to small particles, the ratio of the content of the total REE to molybdenum is ≈1.96, and for spectra 3, 4 (large particles) ≈1.54. With respect to the calculated composition, for which n(REE): n(Mo) ≈1.67, small particles are enriched in rare earth elements, and their composition is close to the formula Ln2MoO6. Thus, small particles can be assigned to a phase with a monoclinic structure, while large ones to a cubic fluorite-like one.Table 3
Chemical element content of the Nd4Tb1Mo3O16+δ sample, at %.
ElementCalcExp. (total)Exp. (small particles, spectra 1, 2)Exp. (large particles, spectra 3, 4)Nd16.316.920.017.2Tb4.14.07.23.4Mo12.212.713.913.4O67.366.458.966.0


## 3.3. Terbium Oxidation State

The value of the substitution limit indicates that terbium exists in the crystal structure of neodymium molybdate predominantly in oxidation state 3+. The chemical determination of the REE content in higher oxidation states is usually carried out using iodometric or permanganometric titration. To confirm the degree of oxidation of terbium in the studied fluorite-like molybdates, a weighed portion of the Nd4.5Tb0.5Mo3O16+δ solid solution was dissolved in sulfuric acid with an excess of manganese sulfate (II), as described in [31]. Under these conditions, Tb4+ ions oxidize manganese (II) to a permanganate ion, which causes a violet color in the solution(8)5Tb4++Mn2++4H2O⟶MnO4−+5Tb3++8H+.We did not observe the appearance of solution color upon dissolution of terbium-containing molybdates; that would prove the existence of terbium in solid solutions only in the 3+ oxidation state.


## 3.4. Rietveld Refinement

The crystal structure refinement of the Nd4.85Tb0.15Mo3O16+δ solid solution was carried out by the Rietveld method using X-ray diffraction data (Figure 7).Figure 7
Experimental (red dots) and calculated (black profile) XRD patterns, their difference (blue profile), and Bragg reflection positions (green lines) of the Nd4.85Tb0.15Mo3O16+δ solid solution.As a result of the calculation (Table4), the predominant location of the substituting element in the Ln2 position was obtained, as in the case of systems with rare earth elements smaller than neodymium [17]. However, the error in the fitted value of site occupancies is much larger than the actual content of terbium in the sample; therefore, such an occupation of lanthanide sites was fixed and was not further refined. The isotropic thermal parameters were refined separately for each position of the cations and in total for all oxygen atoms. Similar to the previous studies [6, 9–11], the value of Biso for atoms in position Ln2 is larger than that of position Ln1. The reliability factors for the last stage of refinement were: RB = 3.03; Rf = 4.68; Rp = 9.35; Rwp = 13.8; Rexp = 10.26; χ2 = 1.81.Table 4
Refined coordinates and isotropic thermal parameters of atoms, as well as the occupancy of positions of Nd4.85Tb0.15Mo3O16+δ solid solution.
АtomxyZBisoOcc.PositionNd10.00939 (2)0.250.250.22 (5)1212eTb10.00939 (2)0.250.250.22 (5)0Nd20001.22 (9)7.48cTb20001.22 (9)0.6Mo00.750.250.53 (7)1212dO10.1213 (11)0.1213 (11)0.1213 (11)1.02 (19)1616fO20.5827 (11)0.3638 (8)0.8291 (10)1.02 (19)4848iThe interatomic distances and bond valence sums (BVS) were calculated based on the refined atomic coordinates (Table5). The introduction of terbium into the neodymium molybdate structure leads to a decrease in the Ln1–O1 and Ln2–O2 distances, while the Ln1–O2 and Ln2–O1 distances slightly increase, which indicates a distortion of the crystal structure upon modification with a smaller ion than neodymium.Table 5
Some interatomic distances and bond valence sums (BVS) for Nd4.85Tb0.15Mo3O16+δ (∗) solid solution compared to Nd5Mo3O16+δ (∗∗).
PositionEnvironmentBVSr (Me– O) (Å)∗r (Me– O) (Å)∗∗Nd14 × O13.09 (4)2.356 (16)2.377 (3)Tb14 × O22.53 (4)2.597 (11)2.567 (3)Nd22 × O12.79 (3)2.320 (15)2.283 (2)Tb26 × O22.29 (3)2.578 (14)2.615 (2)Mo4 × O25.62 (11)1.781 (13)1.768 (2)The suitability of crystallographic positions for accommodating various ions and the stability of crystal structure can be assessed using the bond valence method [32].The BVS of terbium ions is significantly lower than the expected value of +3 in both positions. This result suggests that Tb–O bonds are under tensile stress, giving rise to the instability of the crystal structure. At the same time, the deviation of BVS for cations from the oxidation state for the Ln1 position is smaller than for the Ln2 one, which indicates that the Ln1 position is more suitable for the location of both terbium and neodymium ions. Nevertheless, according to the refinement of the occupancy of the crystallographic positions, the terbium atoms are located in the Ln2 polyhedron. This is probably due to the strong distortion of the Ln2 polyhedron, which is characterized by the shortest Ln2–O1 distances, making it difficult to locate a large Nd3+ ion in this position.


## 3.5. Conducting Properties

Figure8 shows the temperature dependence of the specific conductivity of neodymium molybdate and the Nd4.7Tb0.3Mo3O16+δ solid solution. Low-temperature parts of the dependence of terbium-contained sample can be attributed to impurity conductivity and high-temperature area to intrinsic one.Figure 8
Temperature dependence of conductivity for samples of the Nd5Mo3O16+δ molybdate () and the Nd4.7Tb0.3Mo3O16+δ solid solution ().Doping of neodymium molybdate by terbium leads to decreasing conductivity within the entire temperature range. It is known that the arrangement of ions with a variable oxidation state in equivalent positions should lead to a significant contribution of electronic conductivity [33]. For the modified samples of the Nd5−xTbxMo3O16+δ series, a decrease in conductivity values indicates the existence of terbium only in the +3 oxidation state in solid solutions with a fluorite-like structure.


## 3.6. Atomistic Modeling

The atomic coordinates and interatomic distances calculated as a result of the Nd5Mo3O16+δ molybdate structure optimization are in satisfactory agreement with the experimental values (Table 6). The calculated unit cell parameter is close to the value for neodymium molybdate obtained under reducing conditions [1, 10, 34]. The slightly smaller values of interatomic distances obtained from the simulation results compared with the experimental data are due to the smaller unit cell parameter, which is obviously due to the absence of “overstoichiometric” oxygen in the model.Table 6
Experimental and calculated unit cell parameters and interatomic distances of Nd5Mo3O16+δ crystal structure.
CalcExp∗Exp [9]Exp [34]a (Å)10.972511.03916 (16)11.03379 (9)10.9959 (7)Interatomic distances (Å)Nd1–O1 × 42.3462.377 (3)2.365 (1)2.3612 (17)Nd1–O2 × 42.5562.567 (3)2.578 (1)2.566 (3)Nd2–O1 × 22.2342.283 (2)2.291 (2)2.293 (4)Nd2–O2 × 62.6092.6149 (18)2.608 (1)2.569 (3)Mo–O2 × 41.7711.767 (2)1.776 (1)1.791 (4)∗Present work.The usage of refined values of the Buckingham potential for the Tb3+ ion also makes it possible to satisfactorily describe the crystal structure of terbium molybdates (Table 7). Attention is drawn to the larger values of the unit cell parameters obtained from the full-profile analysis of the X-ray diffraction pattern for Tb5Mo3O16.5 with a monoclinic structure compared to Tb2MoO6 molybdate [26]. Since Tb5Mo3O16.5 molybdate ( composition can be represented as Tb5/3MoO5.5) has the same structure as Tb2MoO6, it contains less terbium than Tb2MoO6, which leads to the formation of vacancies in the terbium and oxygen positions. In this case, one would expect a decrease in the unit cell parameters with a decrease in the terbium content. However, the simulation results show that for the model containing vacancies at the sites of terbium and oxygen, the unit cell parameters are also larger than for Tb2MoO6 molybdate, which agrees satisfactorily with the experimental values. The discrepancy between the calculated parameters of the crystal structure and the experimental ones may be due to the fact that in the model, the vacancies at the sites of terbium and oxygen were statistically distributed between the positions of the crystal structure.Table 7
Experimental and calculated parameters of monoclinic terbium molybdate crystal structure.
Tb5Mo3O16.5 expTb5Mo3O16.5 calcTb2MoO6 [26]Tb2MoO6 calcUnit cell parametersa (Å)16.493 (3)16.557516.4554 (3)16.4748b (Å)11.150 (2)11.092911.1143 (2)11.0421c (Å)5.4060 (12)5.44365.3936 (1)5.3847β (°)108.442 (7)109.79108.428 (1)109.68ρ (g/cm3)6.366.347.247.34Interatomic distances (Å)Tb1–O4 × 22.77 (3)2.9682.770 (5)2.863Tb1–O5 × 22.30 (3)2.3102.319 (5)2.330Tb1–O6 × 22.22 (2)2.2422.223 (5)2.248Tb1–O6 × 22.39 (3)2.4122.387 (5)2.335Tb2–O2 × 22.42 (4)2.3362.377 (6)2.352Tb2–O2 × 22.491 (19)2.5232.488 (4)2.484Tb2–O4 × 22.516 (17)2.3902.478 (5)2.398Tb2–O5 × 22.25 (3)2.3612.238 (5)2.297Tb3–O1 × 12.57 (4)2.5202.499 (5)2.461Tb3–O1 × 12.63 (3)2.5452.570 (5)2.477Tb3–O3 × 12.31 (3)2.4512.426 (6)2.373Tb3–O3 × 12.41 (4)2.4582.435 (5)2.445Tb3–O4 × 12.62 (4)2.8502.655 (5)2.783Tb3–O5 × 12.34 (3)2.2452.265 (6)2.270Tb3–O6 × 12.21 (4)2.1832.213 (6)2.198Tb3–O6 × 12.38 (3)2.4232.273 (5)2.359Mo–O1 × 11.74 (4)1.7681.796 (5)1.765Mo–O2 × 11.74 (4)1.7791.799 (6)1.775Mo–O3 × 11.79 (3)1.8021.812 (6)1.801Mo–O4 × 11.84 (4)1.7971.805 (6)1.796Mo–O5 × 12.15 (4)1.9692.214 (6)2.085Mo–O1 × 13.02 (4)3.1353.113 (5)3.088Mo–O2 × 13.17 (4)3.0813.072 (5)3.027Mo–O3 × 13.01 (4)2.9712.983 (6)2.970The Mott-Littleton method was used to calculate the energies of substitution defects of neodymium with terbium in positions 12e and 8c. The simulation results show that terbium placement in the Ln2 (8c) position of the neodymium molybdate crystal structure is energetically more favorable (Table8).Table 8
Energy of substitutional defect (Ed (NdTb×)) and substitution process according to equation (1) (Es), eV.
PositionLn1Ln2Ed (NdTb×)0.500.43Es1.091.02The quantitative relationship between the probabilities of terbium placement in positions Ln1 (12e) and Ln2 (8c) and the calculated defect energies E1 and E2, respectively, can be determined using the Boltzmann distribution in the form(9)P2P1=eE1−E2/−kT,where P1 and P2 are the probabilities of terbium placing in crystallographic position with defect energies E1 and E2, respectively; T is the synthesis temperature (1323 K), k is the Boltzmann constant (8.617·10–5 eV/K). The resulting ratio of the terbium content in the Ln2 and Ln1 positions calculated by the substitutional defect energies is 1.84.


## 4. Conclusions

The investigation of the Nd5−xTbxMo3O16+δ system revealed that increasing the terbium concentration leads to the phase transformation from a cubic fluorite-like to a monoclinic structure. The homogeneity areas of cubic and monoclinic phases are 0 ≤ x ≤ 0.52 and 1.2 ≤ x ≤ 5, respectively. Increasing terbium content leads to a decrease in unit cell parameters for both molybdates. Tb5Mo3O16+δ molybdate enters the Tb2MoO6 homogeneity region like a subtractive solid solution and is characterized by larger unit cell parameters compared with Tb2MoO6. Atomistic simulation shows a more favorable placement of terbium in the 8c position in the Nd5Mo3O16+δ crystal structure, which was confirmed by Rietveld refinement of solid solutions. The ratio of terbium content in the Ln2 and Ln1 positions is 1.84. The use of the Tb3+ fitted potential leads to good agreement between the experimental and calculated crystal structures of terbium-containing molybdates. Despite the fact that the introduction of terbium did not lead to an increase in the neodymium molybdate conductivity, the obtained results will be useful for the systematic study of isomorphic substitutions and the properties of terbium-containing compounds.



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