Introduction

The sol–gel method is widely used in the synthesis of inorganic substances, ceramic materials and complex oxide systems. The use of this method allows to increase the homogeneity of a material, reduce the temperature of synthesis, and simplify the methods of obtaining the materials.

For the first time the compounds under study in the system BaO–La2O3–WO3 were obtained by the authors [1,2,3], who also determined the crystal structure for BaLa2WO7. The space group P21/b with the lattice parameters: a = 8.857 Å, b = 12.872 Å, c = 5.830 Å, γ = 105.16° was initially determined for this compound [1]. In the subsequent making the structure more precise according to the data of X-ray diffraction of single crystals the space group Pb was chosen; in this case the presence of pseudo-centers of symmetry was noted as well as the fact that a transition into centrosymmetric space groups P2/b and P21/b is not possible [2].

Later, research into the crystal structure of the samples BaLa2WO7 and SrLa2WO7 was carried out using the powder X-ray diffraction and neutron diffraction [4]. As isostructural compounds β-La3RuO7 and La3IrO7 were used. The samples under study were found to be isomorphic to each other and belong to the centrosymmetric space group of P1121/b symmetry.

The capability for luminescence was shown for samples SrLa2WO7 and BaLa2WO7, doped with neodymium, and for BaLa2WO7, doped with europium [5, 6]. The luminescence properties of materials based on BaLa2WO7 doped with different lanthanides have been widely investigated in a number of subsequent studies [7,8,9,10].

It should be noted that in all the above studies the method of solid phase reaction was used for synthesizing compounds at temperatures of 1200–1350 °C. As it is noted in work [6], decomposition of the compound into simpler phases is observed at the temperature of 1500 °C. Therefore, to improve efficiency, the synthesis by the “sol–gel” method [11,12,13,14,15] is carried out at reduced temperatures. This paper presents the results of the synthesis of the compounds with alkali earth elements and investigates their structural and thermodynamic properties.

Experimental

Synthesis

As initial materials for the synthesis, La2O3, Na2WO4·2H2O, MgCl2·6H2O, CaCl2·6H2O, SrCl2·6H2O, and BaCl2·2H2O of not less than 99.8% of purity were used. Lanthanum oxide was additionally purified by dissolving in nitric acid (68%), precipitating oxalic acid, and calcining the obtained lanthanum oxalate at the temperature of 850 °C; the salts were purified by recrystallization.

To obtain precursors by the “sol–gel” method, 6.516 g (0.02 mol) of La2O3 were dissolved in a mixture of citric and nitric acids (Solution 1). We dissolved 6.597 g (0.02 mol) of Na2WO4·2H2O in distilled water and added dropwise to it a solution of MeCl2 under continuous stirring. The precipitate was washed and filtered in a Buchner funnel under vacuum. The obtained freshly precipitated MeWO4 was dissolved in citric and nitric acids (Solution 2). Solution 2 was added to Solution of 1 under constant stirring at 70 °C for 2 h. The mixture of the solutions was evaporated to form a gel.

Synthesis of the complex oxides was conducted by means of a stagewise annealing of the obtained citrate precursors in a muffle furnace Nabertherm N7/H. The decomposition of the citrate precursors was conducted at temperatures of 200–600 °C. A complete burning of carbon particles was achieved at 800 °C. The formative annealing was continued at temperatures of 900, 1000, and 1150 °C to control phase composition of the products. The time of intermediate annealing at each stage was 4 h, and the time of the final annealing was 8 h at 1150 °C. After each stage, the products were triturated in agate mortar.

Measurements

The X-ray diffraction patterns of the synthesized samples were registered at room temperature on a X-ray diffractometer XRD­6000 (Shimadzu). The measurements were carried out over the range of angles 2θ from 20° to 60° with an increment of 0.02° at radiation of CuKα, λ = 1.54056 Å. Phase analysis and indexing of the obtained diffraction patterns was performed using programs Match! v.2.4.7 and FullProf [16]. The unit cell parameters of the synthesized samples have been refined by computer program PowderCell v.2.4.

Studies of the microstructure and composition of the samples were carried out by the scanning electron microscopy (SEM) and electron probe microanalysis (EPMA) on an electron probe micro-analyzer JXA-8230 of JEOL company at an accelerating voltage of 25 kV and an electron beam current of up to 25 nA. For all the areas of the samples, the mode of backscattered electrons (COMPO) was used. The mode gives better quality images of such objects compared to the monitoring mode and imaging in secondary electrons (SEI). The basic data in the SEM-images are magnification, a marker, its length in microns, date and time, as well as the monitoring mode (COMPO or SEI).

EPMA-studies were carried out using the following methods of analysis: energy-dispersive spectroscopy (EDS), as well as wave-dispersive spectroscopy (WDS). EDS was carried out from an area, WDS was semiquantitative, and also WDS-mapping. A beam diameter used for WDS analysis was 100 µm; the size of the analyzed area using EDS analysis was approximately 110 × 100 µm. The time of WDS analysis was an order of magnitude longer than that of EDS analysis; however, WDS analysis allows to distinguish confidently some lines of the characteristic X-ray radiation, which practically coincide in strontium and tungsten (Lα = 1.806 keV for Sr and Mβ = 1.835 keV for W). Mapping of the samples by the elements was carried out in WDS mode at magnification of 2000 ×.

Spectroscopic studies were performed on a Raman scattering spectrometer Triplemate Spex, equipped with a multi-channel detector LN-1340 PB (Princeton Instruments) in the region of 25–1500 cm−1 and on an infrared Fourier spectrometer Vertex 80 in the region of 80–4000 cm−1.

The low-temperature heat capacity of the compounds obtained was measured over the temperature range of 4.2–320 K by adiabatic calorimetry on installation of the company «Termax» (Moscow, Russia) [17]. The weighted portions of about 1–1.5 g were placed in a titanium container (1 cm3), which then was filled with the heat exchange gas helium and sealed with indium foil. Measurement steps varied from 0.3 K at low temperatures up to 3 K at room temperature. Results of the calorimetric measurements were presented by data sets in 200–300 experimental points obtained in several consecutive series.

Results and discussions

Morphology and stability

The results of EMPA study are presented in Appendix A. A1–A4 in Appendix A show SEM-images of the structure of the samples at different magnifications. Dispersity of the powder is determined by the presence of individual particles and a large number of their agglomerates. The main part of the particles has a size of less than 1 µm and a rounded shape. For calcium sample, two types of particles are observed; the most part of agglomerates is so dense that the particles have almost completely intergrown to form rounded holes. In the COMPO mode, all the particles have a single light contrast, which means a high uniformity of the substances. Figures A5–A8 in Appendix A show WDS-spectra and numerical results of their processing. Based on the results of WDS-mapping by the elements, we can make a conclusion about a high homogeneity of distribution of the elements for the studied compounds of tungsten series. In the calcium sample, it can be seen that calcium atoms are presented in composition of the both phases. Figure A9 in Appendix A shows the full EDS-spectra from the area for the synthesized samples. The spectra registered characteristic peaks belonging only to the elements having been used. The ratios of elements in the samples according to the results of a semiquantitative EDS analysis approximately correspond to the gross formula of the compounds.

Crystal structure

It is seen from X-ray diffraction patterns (Fig. 1) that the samples decrease degree of symmetry depending on increasing the atomic number of an alkaline earth element is observed. At first sight, the sample MgLa2WO7 looks like as if it belongs to the cubic system with the unit cell parameters of the order of a = 11.17 Å. However, more detailed indexing shows that the sample structure is pseudo-cubic, and actually the compound relates to orthorhombic syngony with the unit cell parameters which is close to tetragonal cell.

Fig. 1
figure 1

Refinement of crystal structure and unit cell parameters for the samples. From top to bottom: MgLaW, CaLaW, SrLaW, BaLaW. The arrows mark the admixture peaks

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The sample based on calcium, lanthanum and tungsten can be identified as a mixture of two phases—cubic and hexagonal. The cubic phase is related to the compound La6WO12 with some calcium content (sp. gr. F\(\overline{4}\)3m, a = 11.129 Å), the refinement for this phase was based on works [18, 19]. The former reference describes the undoped compound La6WO12, and the latter reference describes the compound doped with certain quantity of calcium ions. The hexagonal phase corresponds to the compound Ca3La2W2O12 [20] described by Smirnov and coworkers [21, 22]. Refinement of the hexagonal phase (sp. gr. R\(\overline{3}\)/c a = 9.7578 Å, c = 55.468 Å) was carried out taking into account crystallographic data for isostructural compounds from the work [23]. So, this sample consist of two phases; the ratio between the cubic and hexagonal phases is approximately 40:60 by the volume.

In the case of SrLa2WO7 and BaLa2WO7, we took as the basis the data from Ref. [4]. The samples synthesized by us are in good conformity with these data; however, there are some discrepancies. Thus, in our diffractogram for SrLa2WO7 there is no peak at 27.7° (2θ), and there are peaks at 29.7° and 31.2°. The first and second peaks in Ref. [4] were attributed to the presence of SrWO4; the admixture in our sample is most likely to be represented by compound which is isostructural to BaSr5(Lu,Ho)2W3O18 [24]. Further, in the diffractogram of our sample of BaLa2WO7 there are additional peaks at 15.7°, 27.3°, 36.0°, and 39.5°, which indicate the presence of admixture of lanthanum hydroxide La(OH)3. However, having refined the unit cell parameters, we found correlations for the other lines in the list of reflections of hkl planes for P1121/b point group of symmetry.

It should be noted that pure lanthanum oxide, when stored in air, depending on conditions, can form a number of substances: hydroxides La(OH)3, LaO(OH), carbonates La2O2CO3, La2(CO3)3·nH2O, mixed compounds La2(OH)4CO3, La2(OH)4CO3·nH2O and others [25,26,27,28,29,30,31]. In addition, the type of crystalline structure of the starting oxide also affects the formation of certain products. In the case of hexagonal lanthanum oxide degradation, lanthanum hydroxide La(OH)3 is formed relatively quickly, and in the case of cubic modification, the process mainly proceeds along the path of LaO(OH) formation. Subsequently, carbonate compounds may form from these substances. As a rule, the most stable La2O2CO3 or amorphous La2(OH)6−2x(CO3)x·nH2O compounds are accumulated as a result. When lanthanum oxide released from the products of high-temperature synthesis is hydrated in air, the number of affecting factors increases. One can make a conclusion that during annealing, some of the lanthanum oxide was released in the form of a high-temperature hexagonal modification, which was subsequently hydrated to lanthanum hydroxide.

The crystal structures of the compounds have been refined to fit the theoretical diffraction profiles. The obtained results of crystal structure determination of the samples are given in Table 1. The atomic positions of elements in the crystal structure refined for our samples are given in Table 2. The experimental data of the X-ray diffraction of the samples are presented in Appendix B. The calculation results are presented in Appendix C.

Table 1 Parameters of the unit cells of the synthesized compounds
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Table 2 The atomic coordinates of the samples
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Figure 1 shows that for a sample with the theoretical formula MgLa2WO7, good agreement was obtained in the framework of the Pbcm space group of the orthorhombic syngony. Reliability factors amounted to Rp = 7.41%, Rwp = 10.61%. The sample CaLaW was smoothed as a mixture of hexagonal Ca3La2W2O12 and cubic La5.2Ca0.4WO11.2 phases; reliability factors amounted to Rp = 8.63%, Rwp = 12.16%. The volume ratio of phases is 62.4% for the hexagonal phase and 37.6% for the cubic one; in mass terms, the phase ratio is 63.4:36.6 mass%. The refinement of the SrLa2WO7 compound, taking into account the presence of the assumed admixture Sr6La2W3O18, gave reliability factors of Rp = 9.44%, Rwp = 13.27%. The volumetric content of the admixture of 1.4 vol% is approximately equal to its mass fraction of 1.3 wt%, that is, the content of the basic substance is 98.7 wt%. According to the refinement results of the crystal structure, there is a noticeable admixture of lanthanum hydroxide La(OH)3 in the BaLa2WO7 compound (Rp = 8.18%, Rwp = 11.18%). The volume ratio of the phases is 81:19 vol% that in terms of mass fractions gives 86.2:13.8 wt%.

In compound MgLa2WO7, coordination polyhedra of lanthanum La1 and La2 atoms are linked through edges into the chains, which when intersecting form a three-dimensional framework. The octahedra of tungsten are linked through vertices and arranged into chains alternating with polyhedra of magnesium atoms. The coordination polyhedra of magnesium atoms are also in an octahedral configuration.

In accordance with Ref. [4], the structure of SrLa2WO7 (as well as BaLa2WO7) compound is formed by the layers of alternate polyhedra of lanthanum La1 and La2 atoms linked together by sharing O–O edges. These layers, in turn, are linked together through the octahedra of tungsten atoms, between which the polyhedra of strontium atoms are located as well. The coordination number in polyhedra of lanthanum and strontium atoms is 8.

By analogy, WO6-octahedra and polyhedra of La1 and La2 atoms with the coordination number 8 are also present in the structure of BaLa2WO7, but the polyhedra of barium have the coordination number 9. This fact is caused, on the one hand, by a larger ion radius of barium than that of strontium, on the other hand, it is conditioned by the localization of the atomic positions of barium in a position relatively more equidistant from the oxygen atom O1, which leads to the formation of an additional coordination.

According to the literature, the compounds with the formula of pyrochlore type A2B2O7 are usually crystallized in the cubic system with a space group symmetry of Fd\(\overline{3}\)m [32,33,34,35] and Fm\(\overline{3}\)m [36,37,38], however, sufficiently great number of compounds has a lower symmetry of the unit cell [39,40,41,42,43,44,45,46]. For the compounds of A2B2O7 type, the stability of the pyrochlore structure is determined by the ratio of ionic radii of cations rA/rB over the range from 1.46 to 1.78 [47, 48]. Changing in the ratio rA/rB toward the smaller side leads to the structure of a defective fluorite; if the ratio is increased, the compound crystallizes in the monoclinic system. In the presence of a partial cationic substitution, an disposition can become more complicated, especially in case of a significant difference between the ionic radii of A and B atoms. As a result of such distortions of the unit cell, pyrochlore with the trigonal [39, 40], tetragonal [41, 42], orthorhombic [43], and monoclinic system [44,45,46] can be formed.

For the compounds under study, this ratio is satisfied only in the case of magnesium, and it is very close to the stability limit. For other alkaline earth elements, a deviation from the boundaries of stability is even greater, and the structures being formed are far from the cubic ones. In the case of calcium, in addition, its ionic radius is very close to the ionic radii of lanthanum and other lanthanides [49,50,51]. Thus, lanthanum can be replaced by calcium in the crystal structure in nonequivalent positions, and in some cases this distribution leads to dividing the structure into separate phases [52,53,54]. Presumably, a similar situation is observed in the case of our calcium sample that gives two compounds. It should be mentioned that in the CaO–Nd2O3–WO3 system the formation of two phases (cubic and hexagonal) was also noted in the sample synthesized [55].

Vibrational spectra

The Raman spectra of the compounds (Fig. 2) can be divided into four frequency bands: (a) 700–1000 cm−1—stretching vibrations of W-polyhedra (most likely, octahedra WO4); the most intense mode in this range refers to the totally symmetric vibration of the octahedron; (b) 400–600 cm−1—deformation vibrations of W-polyhedra; (c) 300–400 cm−1—libration vibrations of W-polyhedra (restricted turns); (d) 50–300 cm−1—mixed translational vibrations of cations and W-ions.

Fig. 2
figure 2

Raman spectra of the samples. From top to bottom: MgLaW, CaLaW, SrLaW, BaLaW

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One can note that in the Raman spectra of the samples there are no noticeable sharp peaks in the area of 1050–1100 cm−1, which correspond to vibrations of the [CO3]2− group [31] that indicates the absence of lanthanum oxycarbonates in the samples obtained. In the spectrum of BaLa2WO7, there are signs indicating the presence of lanthanum hydroxide. The basic vibrational lines active in the Raman spectrum of La(OH)3, which are located at 800, 687, and 589 cm−1 [56] and correspond to rotational and translational vibrations of OH-groups, overlap with vibrations of the crystal structure of the main phase. Nevertheless, the vibrational frequency at 140 cm−1, corresponding to the translational vibrations of lanthanum atoms in hydroxide, can be correlated with the band in the spectrum of the BaLa2WO7 sample at 138 cm−1, which is absent in the Raman spectrum of SrLa2WO7. One can also note the absence of oscillations characteristic of scheelite structures in the spectra of samples with the participation of calcium, strontium, and barium. The most intense vibrational line of W–O bonds in tetrahedra, located in the range from 910 cm−1 for calcium tungstate up to 925 cm−1 for barium tungstate does not appear in the spectra; also, the second most intense band is not observed at 330 cm−1 [57, 58]. In the case of magnesium, the main peak of MgWO4 at 915 cm−1 [59] would overlap with a wide band at 800–1000 cm−1 associated with the structural features of the sample; however, an X-ray diffraction pattern distinctly shows the absence of magnesium tungstate.

In the spectra of all the samples in the range over 750–800 cm−1, there is a band corresponding to stretching vibrations of W−O bonds in the octahedral configuration, which usually have a length of the order of 1.9 Å [58]. The position of this band shifts somewhat toward longer waves as the mass of alkaline earth metal increases that reflects oscillations frequency reduction of the interconnected oscillators. This peak is rather narrow for SrLa2WO7 and BaLa2WO7 since the bond lengths in the coordination tungsten octahedra are close in size to each other (from 1.90 to 1.98 Å) and oscillate at approximately the same frequency. In the case of calcium, the sample is represented by two compounds: In the cubic phase, the length of the W–O bonds in the octahedra is 1.93 Å that corresponds to a peak at 780 cm−1, while in the hexagonal phase the lengths of the bonds in the octahedra are 1.80–1.83 Å and give peaks at 805–825 cm−1. In the spectrum, these bands overlap each other and the main peak is wider than that of strontium and barium. Further, in the spectrum of a magnesium sample appear bands that at first glance corresponding to both vibrations of W–O bonds in octahedra (peaks from 780 to 850 cm−1) and vibrations of bonds in WO4 tetrahedra (the band with a maximum at 900 cm−1) [58]. Nevertheless, this circumstance is primarily due to the very different bond lengths in the distorted octahedra of the crystal structure of the sample. As a result, the ensemble of shorter bonds in the polyhedron oscillates as a single fragment. A similar picture is observed in the spectra of other compounds with a significant distortion of tungsten–oxygen octahedra, for example, in CsLuW2O8 [60]. In addition, in the structure of the sample the possibility of a partial replacement of other cations with tungsten in their crystallographic positions to form a tetrahedral environment is not ruled out. A similar combination of tetrahedral and octahedral groups in the structure of compounds is observed, for example, in sodium ditungstate Na2W2O7 [61, 62] and in solid solutions Bi2−xLaxWO6 [63].

In the area of deformation and libration vibrations, the samples also have maxima corresponding to the octahedral environment of tungsten atoms. They relate to bending and flapping vibrations of O−W−O bond angles. For the calcium-, strontium-, and barium-based samples, these bands are grouped in the area of 250–300 cm−1 and 400–450 cm−1 into fairly sharp peaks, showing relatively high symmetry of the octahedra. In a magnesium sample, these vibrations are represented by relatively wide bands that reflects frequency variations in the distorted structure of the polyhedra.

In translational vibrations region, in the sample MgLa2WO7 the vibration frequency of light magnesium is quite high (200–300 cm−1) and it is partially overlapped by the region of libration vibrations of octahedra. In the sample, BaLa2WO7, vibrations of heavy Ba are shifted to the low-frequency region of 50–150 cm−1. Basically, the spectra of SrLa2WO7 and BaLa2WO7 are similar due to the identity of the crystal structure of the samples.

The infrared spectra of all the compounds (Fig. 3) are represented by broad bands, but a width of the lines is not necessarily determined by disordering, but it is associated with physical peculiarities of the IR-absorption spectra occurrence. As can be seen from the figure, the essential vibrations of the compounds studied are located in the region of wave numbers up to 1000 cm−1, which correspond to interatomic vibrations in the crystal lattice of the substances.

Fig. 3
figure 3

IR-spectra of the samples. From top to bottom: MgLaW, CaLaW, SrLaW, BaLaW

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The peaks observed in the region of 500–1000 cm−1 correspond to symmetric and antisymmetric longitudinal vibrations of the valence bonds of Me−O, La−O and W−O. The deformation vibrations of the valence bonds are located in the region of up to 500 cm−1 and they are partially observed in the presented spectra. The basic vibration frequency for a tetrahedral anion \({\text{WO}}_{{4}}^{{{2} - }}\) is 931 cm−1 (stretching symmetric ν1), 373 cm−1 (deformation symmetrical ν2), 833 cm−1 (stretching antisymmetric ν3) and 320 cm−1 (deformation antisymmetric ν4) [64, 65]. In the spectra of the synthesized samples, these frequencies correspond to the vibrations of tungsten–oxygen bonds in the octahedral environment and they are shifted to the long-wave area. Thus, for the calcium sample vibrations of W−O bonds are found at 720–760 cm−1, 350–370 cm−1, 600–650 cm−1 and 260–310 cm−1, while the bands of the vibrations in the structure of scheelite CaWO4 are identified at 910 cm−1, 408 cm−1, 823, and 769 cm−1, 330, and 290 cm−1 [66]. A similar picture is observed also for other tungstates of alkaline earth metals that confirm an absence of scheelite admixtures in the samples [66,67,68].

Free vibrations of the bond La−O are placed at 812 cm−1; for alkaline earth elements free vibrations of the bond Me−O are, respectively, 785 cm−1, 650 cm−1, 653 cm−1, and 670 cm−1 in order of increasing the atomic number of an element [65]. In the spectra of the compounds obtained, these vibrations correspond to the bands at 790–880 cm−1, 570 cm−1, 550 cm−1, 500 cm−1, and 510 cm−1. In general, the pattern of band shift toward the high-frequency region with decreasing atomic weight of an alkaline earth element is analogous to that observed in similar compounds with increasing the proportion of a lighter element [69]. At the same time, the absorption bands show no vibrations in the region characteristic for the bridge groups W–O–W, suggesting that there are no bridged groups of anions in the form of W2O7 in the crystal structure of the magnesium and calcium samples [70]. A weak band of 3400–3600 cm−1 refers to the vibrations of H2O in the matrix of KBr, in which the powder is pressed.

Heat capacity and thermodynamic function.

The measured dependences of the heat capacity of the samples over the range from temperature of liquid helium to room temperature showed no deviations from the normal course of the curve. The experimental data were processed by the spline-approximation method. The values of the coefficients of polynomials of the third degree have been obtained and thermodynamic functions of the substances have been calculated over the range of 5–320 K. The temperature dependences of the heat capacity of the samples studied are presented in Fig. 4 and in Appendix D.

Fig. 4
figure 4

Molar heat capacity of the samples. From top to bottom: MgLa2WO7, SrLa2WO7, BaLa2WO7

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For SrLa2WO7 and BaLa2WO7 samples, the experimental heat capacity was corrected for the presence of admixtures detected. For BaLa2WO7, the specific heat capacity of lanthanum hydroxide La(OH)3 [71] was subtracted from the total experimentally measured specific heat capacity of the sample proportionally to the admixture mass fraction. In the case of SrLa2WO7, due to the lack of direct experimental measurements of the heat capacity of the substances corresponding to the formula for the detected admixture Sr6La2W3O18, the known heat capacities for strontium oxide SrO [72], lanthanum oxide La2O3 [73], and strontium tungstate SrWO4 [74] were used to make corrections. The heat capacities of these substances were summed up based on the representation of the formula as 3SrO·La2O3·3SrWO4, after which a correction was made for this heat capacity proportionally to the admixture mass fraction. It should be noted that in Ref. [74], the theoretical calculation of the specific heat of tungstates of alkaline earth metals MeWO4 that showed good agreement with the experimental measurements of CaWO4 [75] and BaWO4 [76] was carried out. The reliability of the introduced corrections is confirmed by proximity of the corrected values of the heat capacity of BaLa2WO7 to the heat capacities of Ba(La1−xPrx)2WO7 compounds with doping from 0.5 to 5 at.% [77].

Extrapolation of the data to absolute zero for the samples was done using an odd-degree polynomial in the form of Cp = aT3 + bT5. The data on the measurements of the heat capacity of the samples are processed by polynomials of the third degree of the form Cp,m = a0 + a1T + a2T2 + a3T3 to get the smoothed dependences. The thermodynamic functions of the compounds in the range of 5–320 K are determined on the basis of the coefficients obtained in the equations of the heat capacity polynomials in accordance with the formulas.

$$\begin{aligned} & S_{{\text{m}}}^{^\circ } (T) = \int\limits_{0}^{T} {\frac{{C_{{{\text{p}},{\text{m}}}} }}{T}{\text{d}}T} \\ & H_{{\text{m}}}^{^\circ } (T) - H_{{\text{m}}}^{^\circ } (0) = \int\limits_{0}^{T} {C_{{{\text{p}},{\text{m}}}} {\text{d}}T} \\ & \Phi_{{\text{m}}} (T) = S_{{\text{m}}}^{^\circ } (T) - \frac{{H_{{\text{m}}}^{^\circ } \,(T) - H_{{\text{m}}}^{^\circ } (0)}}{T}. \\ \end{aligned}$$

Relative deviations for the smoothed thermodynamic functions were estimated on the base of the 95% confidence interval for scattering of the experimental points (Appendix E).

From the low-temperature heat capacity of the samples measured, the characteristic temperatures of the compounds were obtained in accordance with the equation Cp(T) = 2.4π4nR(T/θD)3 [78]. In the case of a magnesium-based sample, the θD value is 319 K, for a strontium sample it equals to 302 K, and for a barium sample it is 278 K. It can be seen that the characteristic temperature decreases with increasing the atomic mass of the alkaline earth element that reflects decreasing in the interatomic interaction in the crystal structure of the compounds.

The standard values of the thermodynamic functions—the entropy S(T), the enthalpy change H(T) − H(0) and the reduced thermodynamic potential Φ(T)—are presented in Table 3. These values are characterized by a smooth increase over the entire measured temperature range without any signs of irregular behavior. The values of relative errors at standard temperature for Cp, S° and H(298.15) − H(0) are within the range of 0.31–0.59% for MgLa2WO7; 0.41–0.62% for SrLa2WO7 and 0.37–0.58% for BaLa2WO7.

Table 3 The thermodynamic functions of the samples
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Conclusions

As a result of citrate synthesis by the “sol–gel” method, the compounds of the series MeLa2WO7 (Me = Mg, Sr, Ba) were synthesized. The study of their structural parameters showed a gradual distortion of the compounds crystal structure from the nearly cubic to monoclinic system when changing an alkaline earth cation from Mg to Ba. In the case of a calcium sample, cation dissipation at crystallographic positions is added to the distortions that leads to dividing the structure into two phases. The results of spectroscopic studies confirm the presence of coordination tungsten octahedra in the structure of compounds. The adiabatic calorimetry of the compounds showed the absence of abnormal signs of phase transitions in the samples. The heat capacity below 20 K submits quite well to the low-temperature approximation of the Debye law. The dependences of the thermodynamic functions of the compounds have a usual course and continuously increase over the temperature range studied.