A. Synthesis

Polycrystalline powders of the BaLa₂Cu_1−x Ba _xTi₂O₉ (x = 0.00, 0.15, and 0.30) ceramics were prepared using a solid-state reaction route in air, and the raw materials BaCO₃ (99.98%), La₂O₃ (99.999%), CuO (99.99%), and TiO₂ (99.8%) were used as starting reagents (all received from Sigma-Aldrich). The quantity of each reagent was weighed according to its stoichiometric coefficient to obtain the appropriate metal ratios of the final products, then mixed and ground in an agate mortar to form a homogeneous powder. The resulting mixtures were placed in alumina crucibles and then increasingly heated in air at different temperature stages with intermittent grinding; these three samples were calcined at 1,000 °C for 20 h and 1,200 °C for 12 h, respectively. During the heat treatment process, the powders were cooled to room temperature, reground, and sintered several times to improve homogeneity. The chemical reaction is as follows:

$$ \begin{array}{c}\left(1+x\right)\;{\mathrm{Ba}\mathrm{C}\mathrm{O}}_3+{\mathrm{La}}_2{\mathrm{O}}_3+\left(1\hbox{--} x\right)\;\mathrm{C}\mathrm{uO}+2\;{\mathrm{Ti}\mathrm{O}}_2\\ {}\frac{\Delta \left(1200\;{}^{\circ}\mathrm{C}\right)}{\mathrm{AIR}}\to {\mathrm{Ba}\mathrm{La}}_2{\mathrm{Cu}}_{1\hbox{--} x}{\mathrm{Ba}}_x{\mathrm{Ti}}_2{\mathrm{O}}_9+\left(1+x\right)\;{\mathrm{CO}}_2\\ {}\left(x=0.00,\hskip0.3em ,0.15,\hskip0.3em \mathrm{and}\hskip0.35em 0.30\right)\end{array} $$

B. X-ray powder diffraction

Diffraction data of the BaLa₂Cu_1−x Ba _xTi₂O₉ (x = 0.00, 0.15, and 0.30) ceramics were collected at room temperature on a D2 PHASER diffractometer, with the Bragg–Brentano geometry, using a copper anti-cathode tube as the radiation source (K _α1, K _α2) of the wavelengths λ(K _α1) = 1.54056 Å and λ(K _α2) = 1.54439 Å with 30 kV and 10 mA, Soller slits of 0.02 rad on incident and diffracted beams; divergence slit of 0.5°; antiscatter slit of 1°; receiving slit of 0.1 mm; with sample spinner, and a Lynxeye detector type with a maximum global count rate > 1000,000,000 cps. The XRD patterns were scanned in steps of 0.020287° (2θ), between 15° and 105° (2θ) with a fixed-time counting of 2 s per step. The WinPlotr software (Roisnel and Rodríguez-Carvajal, Reference Roisnel and Rodríguez-Carvajal2001), integrated into the FullProf program (Rodríguez-Carvajal, Reference Rodríguez-Carvajal1990), was used to analyze the crystal structures by means of the Rietveld method (Rietveld, Reference Rietveld1969). The peak shape was described by a pseudo-Voigt function using linear interpolation between set background points with refinable heights.

The following structural and instrumental parameters – scale factor, unit-cell parameters, pseudo-Voigt corrected for asymmetry parameters, full width at half maximum (FWHM) parameters (u, v, and w), preferred orientation, atomic positions, and isotropic displacement parameters – were included in the refinement of the investigated XRD patterns.

A. Crystallite size and strain

The crystallite size of the three studied compositions (x = 0.00, 0.15, and 0.30) was determined using X-ray line broadening analysis via the Scherrer equation and the Williamson–Hall (W-H) method. X-ray peak broadening can result from crystallite size, microstrain, instrumental effects, temperature factors, and solid solution inhomogeneity. The Scherrer equation is expressed as follows (Scherrer, Reference Scherrer1918):

(1) $$ D=\frac{K\lambda}{\beta_{(s)}\cos \theta }, $$

where D is the average crystallite size (nm), K is the Scherrer constant equal to 0.9, λ is the X-ray wavelength (Cu-K _α average ≈ 1.54178 Å), θ is the Bragg angle corresponding to the most intense XRD peak, and β _(s) is the FWHM of the highest peak. In this study, the crystallite size was deduced from the XRD patterns using the strongest peak located at approximately 2θ ≈ 31.89°. The average crystallite size calculated using the Debye–Scherrer equation was found to be 113 nm (x = 0.00), 100 nm (x = 0.15), and 96 nm (x = 0.30).

The broadening caused by microstrain, which arises from lattice distortions and imperfections, was analyzed based on the relation originally introduced by Stokes and Wilson (Reference Stokes and Wilson1944), who described the strain-induced peak broadening as

(2) $$ \varepsilon =\frac{\beta_{(D)}}{4\tan \theta}, $$

where ε ≈ Δd/d represents the lattice strain and β _(D) is the broadening due to deformation. Williamson and Hall (Reference Williamson and Hall1953) combined this strain broadening equation with the Scherrer equation under the assumption that both size and strain broadening contributions are independent and follow Lorentzian (Cauchy) profiles. This combination leads to the following expression:

(3) $$ {\beta}_{hkl}={\beta}_{(D)}+{\beta}_{(s)}, $$

and from Eqs. (1) and (2), we get

(4) $$ {\beta}_{hkl}\cos \theta =\varepsilon\;\left(4\sin \theta \right)+\frac{K\lambda}{D}. $$

Plotting β _hkl cosθ versus 4sinθ (as shown in Figure 1) allows direct determination of the crystallite size from the y-intercept (Kλ/D) and the microstrain (ε) from the slope of the linear fit. The crystallite size and strain for the three compositions are summarized in Table I. It can be noted that the crystallite size values obtained from the W-H method are larger than those derived from the Scherrer equation alone.

Figure 1. Williamson–Hall plots of the BaLa₂Cu_1−x Ba _xTi₂O₉ phases.

TABLE I. Rietveld refinement details, crystallite size, and strain for the BaLa₂Cu_1−x Ba _xTi₂O₉ phases.

Abbreviation: FWHM, full width at half maximum.

B. Crystal structure analysis and structure description

At room temperature, the PXRD patterns of the compositions BaLa₂Cu_1−x Ba _xTi₂O₉ (x = 0.00, 0.15, and 0.30) are successfully identified by the PDF2 database (Gates-Rector and Blanton, Reference Gates-Rector and Blanton2019) integrated into the HighScore plus software (Degen et al., Reference Degen, Sadki, Bron, König and Nénert2014) and analyzed using the DICVOL program (Louër and Boultif, Reference Louër and Boultif2014) to adopt a phase related to a simple perovskite structure.

A minor presence of secondary phases was detected in the XRD patterns of the as-synthesized samples with compositions x = 0.00 and x = 0.30. For x = 0.00, a small impurity phase was identified as La₂Ti₂O₇ (PDF: 00-027-1182), which crystallizes in the monoclinic system (space group P2₁/m) with lattice parameters a = 13.01 Å, b = 5.54 Å, c = 7.80 Å, and β = 98.37°. For x = 0.30, the detected secondary phases included La₂BaO _x (PDF: 00-042-0337), BaLa₂O₄ (PDF: 00-042-1500), and CuLa₂O₄ (PDF: 01-081-2446).

Figure 2 presents the powder XRD patterns of BaLa₂Cu_1−x Ba _xTi₂O₉ (x = 0.00, 0.15, and 0.30) ceramics collected at 298 K. From the enlarged peaks in the inset of Figure 2, the XRD patterns of the powdered samples show an obvious shift of the peaks toward the lower angles (2θ, Bragg position), indicating that a partial substitution of Cu²⁺ by Ba²⁺ can have an influence on the crystal structure parameters. The unchanged profile of the broadened peaks implies structural stability without evidence of a phase transition. The crystal structures of all studied compositions were analyzed using the Rietveld refinement method (Rietveld, Reference Rietveld1969). Previously, the BaLa₂CuTi₂O₉ structure (Iturbe-Zabalo et al., Reference Iturbe-Zabalo, Igartua, Aatiq and Pomjakushin2013) has been reported as crystallizing in the tetragonal symmetry and space group I4/mcm (No. 140); its atomic coordinates and lattice parameters were used as starting data for refinement.

Figure 2. X-ray diffraction patterns of BaLa₂Cu_1−x Ba _xTi₂O₉ perovskites collected at room temperature. The inset shows an enlarged view of the peaks. The stars (*) for x = 0.00 and 0.30 indicate the presence of impurities.

Table I gives the crystal data, data collection, and details of the Rietveld refinements, including lattice parameters, unit-cell volume, crystal system, space group, and various statistical agreement factors. The values of the reliability factors (%), R_B, R_F, R_p, R _wp, and R _exp, as well as the goodness-of-fit (χ ²), are small and indicate that we have obtained a good quality of the refinement. The quality of the fits is also confirmed by Figure 3, which presents excellent agreement between the experimental (dots) and theoretical (solid line) patterns. The criteria for refined structures are met, ensuring that they are both chemically plausible and statistically reliable. The structural illustration of the tetragonal phase with space group I4/mcm is shown in Figure 4; it was drawn using the Vesta software (Momma and Izumi, Reference Momma and Izumi2011).

Figure 3. Final Rietveld refinement plots for the tetragonal BaLa₂Cu_1−x Ba _xTi₂O₉ phases. The three experimental datasets are represented by red dots, and the calculated patterns are shown as black solid lines. The green vertical ticks indicate the Bragg positions of the main phase. The blue line at the bottom represents the difference between the observed and calculated patterns.

Figure 4. Structural view of the tetragonal phase Ba_1/3La_2/3Cu_1/3Ti_2/3O₃ (x = 0.00) with space group I4/mcm. Panel (a) displays a three-dimensional perspective of the crystal lattice illustrating the arrangement of the Ba/La atoms (large yellow spheres), Cu/Ti atoms (blue spheres), and oxygen atoms (small red spheres) within corner-sharing octahedra. Panel (b) presents the projecti on of the structure along the a-axis, highlighting the layered arrangement and connectivity of the octahedra and the positioning of Ba/La cations in the lattice. Panel (c) illustrates the octahedral tilting consistent with Glazer’s notation a ⁰a ⁰c ⁻.

Rietveld refinement analysis reveals that the BaLa₂Cu_1−x Ba _xTi₂O₉ (x = 0.00, 0.15, and 0.30) structures adopt the tetragonal symmetry of space group I4/mcm (No. 140) with lattice parameters: for x = 0.00, a = 5.6093(1) Å and c = 7.9277(2) Å; for x = 0.15, a = 5.6093(1) Å and c = 7.9269(2) Å; and for x = 0.30, a = 5.6139(2) Å and c = 7.9341(3) Å. The unit-cell volumes were calculated to be 249.43 Å³ (for x = 0.00), 249.42 Å³ (for x = 0.15), and 250.05 Å³ (for x = 0.30). The number of unit formulas within a cell is Z = 4. Reflections from the tetragonal structure with space group I4/mcm (No. 140) are correctly indexed on the studied XRD patterns; where the reflection conditions can be generally expressed as follows: hkl: h + k + l = 2n; hk0: h + k = 2n; 0kl: k, l = 2n; hhl: l = 2n; 00l: l = 2n; h00: h = 2n (International Tables for Crystallography – Volume A). The fractional atomic coordinates and isotropic displacement parameters are listed in Table II. The occupancy factors are given according to the stoichiometry of the formula: (Ba²⁺_1/3La³⁺_2/3)_4b (Cu²⁺_(1-x)/3Ba²⁺ _x/3Ti⁴⁺_2/3)_4c O₃, which is electrically neutral. The Cu²⁺, Ba²⁺(2), and Ti⁴⁺ cations were found to occupy octahedral environments at 4c (0, 0, 0) positions. They are surrounded by six oxygen anions; two O²⁻(1) at 4a (0, 0, 1/4), and four O²⁻(2) at 8h (x, x + 1/2, 0) sites. Whereas the Ba²⁺(1) and La³⁺ cations occupy polyhedral environments at 4b (0, 1/2, 1/4) positions, they are coordinated with 12 oxygen anions; 4 O²⁻(1) at 4a (0, 0, 1/4), and 8 O²⁻(2) at 8h (x, x + 1/2, 0) sites. Despite some differences in ionic size and valence charge, the Cu²⁺(Ba²⁺) and Ti⁴⁺ cations have only one crystallographic B-site. Similarly, the two cations Ba²⁺ and La³⁺ have only one crystallographic A-site, like the simple perovskite structure ABO₃. The isotropic displacement parameters U _iso (Ų) for oxygen atoms at sites 4a and 8h were refined by considering them equivalent; similarly, the Ba(1)/La and Cu/Ba(2)/Ti atoms were considered equivalent at their sites 4a and 4b, respectively. In the composition with x = 0.30, the U _iso values for Ba²⁺(1)/La³⁺ cations were fixed at 0.013 Å² to guarantee the convergence of the refinement.

TABLE II. Atomic positions and isotropic displacement parameters for the tetragonal BaLa₂Cu_1−x Ba _xTi₂O₉ phases.

* Fixed in 0.013 Å² to guarantee the convergence of the refinement.

The selected interatomic distances (Å) and angles (degrees) are given in Table III. The average distances <Ba1/La–O> and <Cu(Ba2)/Ti–O> obtained after the Rietveld refinement of the studied compositions are roughly identical to those found in other perovskite compounds containing Ba²⁺, La³⁺, Cu²⁺, and Ti⁴⁺ cations, such as BaLa₂CuTi₂O₉ with Ba/La–O = 2.804 Å and Cu/Ti–O = 1.99 Å; BaNd₂CuTi₂O₉ with Ba/Nd–O = 2.794 Å and Cu/Ti–O = 1.99 Å; BaPr₂CuTi₂O₉ with Ba/Pr–O = 2.796 Å and Cu/Ti–O = 1.99 Å (Iturbe-Zabalo et al., Reference Iturbe-Zabalo, Igartua, Aatiq and Pomjakushin2013); La₂CuTiO₆ with Cu/Ti–O = 2.02 Å (Gomez-Romero et al., Reference Gomez-Romero, lacin, Casan, Fuertes and Martinez1993); and BaLaCrMoO₆ with Ba/La–O = 2.8402 Å (Musa-Saad, Reference Musa-Saad2014). They are also very close to those calculated from Shannon’s effective ionic radii: Ba²⁺ (1.61 Å), La³⁺ (1.36 Å), Cu²⁺ (0.73 Å), Ti⁴⁺ (0.605 Å), and O²⁻ (1.40 Å) for 12- and 6-fold coordination (Shannon, Reference Shannon1976). The bond M–O2–M angles (M = Cu(Ba2)/Ti) were calculated to be 168.6(7)° for (x = 0.00), 168.6(6)° for (x = 0.15), and 166.8(6)° for (x = 0.30). They are almost identical to the Cu/Ti–O2–Cu/Ti = 167.2(3)° angle observed in BaLa₂CuTi₂O₉ (Iturbe-Zabalo et al., Reference Iturbe-Zabalo, Igartua, Aatiq and Pomjakushin2013). The structural view of the tetragonal compositions BaLa₂Cu_1−x Ba _xTi₂O₉ (x = 0.00, 0.15, and 0.30) with space group I4/mcm (No. 140) is made up of octahedra with shared corners in three dimensions (3D), as shown in Figure 4a. Through the c-axis, the MO₆ octahedra (M = Cu(Ba2)/Ti) are linked via oxygen atoms O(2) of 4a (0, 0, 1/4) positions, and in the ab-plane, they are connected by O(1) atoms of 8h (x, x + 1/2, 0) positions. Due to the special positions occupied by the Cu(Ba2)/Ti atoms at 4c (0, 0, 0) sites, and oxygen atoms O(1) at 4a (0, 0, 1/4) sites, along the c-axis, the bond angle of (M–O1–M) is constrained to be 180°. The volumes of the MO₆ octahedra (M = Cu(Ba2)/Ti) were calculated to be 10.4970 Å³ for (x = 0.00), 10.4954 Å³ for (x = 0.15), and 10.5590 Å³ for (x = 0.30). The volumes of the AO₁₂ polyhedra (A = Ba/La) were calculated to be 51.8616 Å³ for (x = 0.00), 51.8585 Å³ for (x = 0.15), and 51.9539 Å³ for (x = 0.30). Each MO₆ octahedron consists of six bond lengths (2× M–O1) and (4× M–O2) with an average distance of 1.9893 Å for (x = 0.00), 1.9892 Å for (x = 0.15), and 1.9933 Å for (x = 0.30), whereas each AO₁₂ polyhedron consists of 12 bond lengths (4× A–O1) and (8× A–O2) with an average distance of 2.8064 Å for (x = 0.00), 2.8063 Å for (x = 0.15), and 2.8095 Å for (x = 0.30).

TABLE III. Selected interatomic distances (Å) and bond angles (°) for the tetragonal BaLa₂Cu_1−x Ba _xTi₂O₉ phases.

A drawing to visualize the anti-phase tilting of the simple perovskite structure Ba_1/3La_2/3Cu_1/3Ti_2/3O₃ (x = 0.00) is illustrated in Figure 4c; it contains Cu/TiO₆ octahedra tilting in out-of-phase along the [001] _p pseudo-cubic direction in the basal ab-plane. The tilt system follows the Glazer notation a ⁰a ⁰c ⁻ (Glazer, Reference Glazer1972), as derived by Howard and Stokes (Reference Howard and Stokes1998) for simple perovskite structures crystallizing in tetragonal symmetry (space group I4/mcm). The title compositions can exhibit interesting features, such as electrical and photocatalytic properties.