Effect of the ionic radius on structural properties of orthochromites RCrO3 (R = La, Gd, Y)

The main objective of this work was to detail out how to obtain important parameters in the structural analysis of materials with perovskite structure in order to help beginning researchers in this area. In particular, a thorough comparative and investigative study of the effect of ionic radius on the structural properties of orthochromites RCrO3 (R = La, Gd, Y) were presented. It is observed that the b and c lattice parameters increased, whereas a lattice parameter decreased as the ionic radius increased. Consequently, an increase in the unit-cell volume and tolerance factor was observed. The angles and bond lengths increased with an increase in the ionic radius, albeit the inclination angle, as well as the rotation angle, tends to decrease. The distortion parameter suffered a fluctuation in its value according to the increasing ionic radius. Lastly, Williamson-Hall (W-H) analysis caused a decrement in the strain according to decreasing ionic radius.

orthorhombic perovskite unit cell with A-site cation at the corners and B-site cation at the center surrounded by six oxygen atoms located at the face centers to form the oxygen octahedral. In this work, A-site accommodates La, Gd and Y elements, and B-site the Cr element. The rotation of oxygen octahedral about the three cubic axes (100), (010), and (111) by the angles θ, ϕ, and φ, respectively, can accommodate the deviations from this basic structure (Warshi et al., 2018), as shown in Figure 1(a). The orthochromites RCrO 3 (R = La, Gd, and Y) have an orthorhombic symmetry with space group 62 (Pnma) which is a pseudo-cubic space group, and the unit cell parameters are related to the ideal cubic perovskites as ≅ √2 , ≅ 2 and ≅ √2 , at where is the cubic perovskite cell parameter, with the Wyckoff positions of atoms being: R: 2a (x,1/4,z), Cr: 4b (0,0,1/2), O(1): 4c (x,1/4,z) and O(2): 8d (x,y,z) (Romero, Escamilla, Marquina, & Gómez, 2015). orthochromites RCrO3 which has the structure of orthorhombic, with space group 62 (Pnma).
An extremely used parameter to quantify the stability and distortions of perovskite structures, as well as orthochromites, is the Goldschmidt tolerance factor (t) (Bhalla, Guo, & Roy, 2000;Goldschmidt, 1926), defined as: where r R is the radius of the R-cation (valence 3+), r Cr is the radius of the Cr-cation (valence 3+) and r O is the radius of the oxygen in six-coordination (valence -2). If t = 1, we have an ideal cubic perovskite, if we have between 0.75 < t < 1.0 we have other structures such as the orthorhombic, however, if t > 1, it ceases to be perovskite. In the orthochromites, the distortion mechanism is a tilting of rigid CrO 6 octahedron, as can be seen in Figure Another quantity used to quantify the distortion of the octahedron CrO 6 is the distortion parameter ∆ (Romero et al., 2015), which is defined as: where 〈 − 〉 is an average bond length. Finally, using the XRD data, it was possible to estimate the Strain (ε) of the crystalline cell through the Williamson-Hall (W-H) method, which takes into account the distances at half height of the diffraction peaks (Rosenberg et al., 2000). W-H analysis proposes a diffraction line broadening due to crystallite size and strain contribution as a function of diffraction angle which can be written in the form of mathematical expression as β hkl = β t +βε where β t is due to crystallite size contribution, βε is due to strain-induced broadening and β hkl is the full-width at half of the maximum intensity (FWHM) of instrumental corrected broadening (Dhahri, Dhahri, Dhahri, Taibi, & Hlil, 2017). Crystallite size contribution is calculated using the Scherrer equation: The k constant depends upon the shape of the crystallite size (k = 0.9, assuming the circular grain), β t is Full Width at Half Maximum (FWHM) of intensity vs. 2θ profile, λ is the wavelength of the Cu-K ⍺ radiation (λ=1.5406 Å) and θ is the Bragg diffraction angle. While the strain contribution is calculated by: where ε is the strain. We can see clearly that line broadening is a combination of crystallite size and strain, which is represented by the equation: This gives a linear relationship between βcos(θ) and 4sin(θ) when plotted as βcos(θ) (y-axis) vs. 4sin(θ) (x-axis). The crystallite size (D) can be calculated, by fitting Eq. (7), from the interception of the linear fit, and the value of Strain (ε) is calculated from the slope, which is also our interest in this work (Zak, Majid, Abrishami, & Yousefi, 2011).

Results and discussion
As previously mentioned, the values of the parameters used in this work were taken from the ICSD (793440), (251088) and (251108), for the samples LaCrO 3 , GdCrO 3, and YCrO 3 , respectively. The lattice parameters (a,b,c), unit cell volume (V) and the Goldschmidt tolerance factor (t), using the ionic radius of the 3+ = 1.03Å, 3+ = 0.96Å and 3+ = 0.89Å, are shown in Table 1. The atomic positions are shown in Table 2, while the distortion parameter (∆ ) and the inclination (θ) and rotation (ϕ) angles of the octahedrons are shown in Table 3, for the orthochromites RCrO 3 (R = La, Gd, and Y). Table 1. Lattice parameters (a,b,c), unit cell volume (V), the Goldschmidt tolerance factor (t ), and R 3+ ionic radius values (Shannon, 1976).   Figure 3 shows the graph of the lattice parameters (a,b,c), unit cell volume (V) and tolerance factor (t) as a function of the ionic radius (R 3+ = La 3+ , Gd 3+ and Y 3+ ), data from Table 1. The b and c lattice parameters increased whereas a lattice parameter decreased, as the ionic radius of R 3+ increased, as a consequence, an observation in an increased unit-cell volume. Also, the increase of the unit cell volume induces a Goldschmidt tolerance factor increase. Figure 4 shows the graph of geometrical parameters that characterize the orthochromit es as the 〈 − (1) − 〉 and 〈 − (2) − 〉 bond angles, distortion parameter ( ∆ ), inclination (θ), and rotation (ϕ) angles of the octahedrons, as a function of the ionic radius (R 3+ = La 3+ , Gd 3+ , and Y 3+ ), data from Table 3. As can be seen, the 〈 − (1) − 〉 and 〈 − (2) − 〉 bond angles increase with an increase in the ionic radius, and wherefore the angle of inclination (θ), as well as rotation cos(Φ), tends to decrease with an increase in the ionic radius because they are inversely proportional. The distortion parameter (∆ ) suffers fluctuations in its values according to the increasing ionic radius, and this happens because it takes into consideration the same − (1), − (2) and − (2): 1 bond lengths, as well as the same 〈 − 〉 average between them. Figure 5 shows the adjustment of Williamson-Hall (W-H) using equation (7), which is plotted as a function of β hkl cos(θ) (y-axis) vs. 4sin(θ) (x-axis) for the LaCrO 3 , GdCrO 3, and YCrO 3 structures. With the linear adjustment, it is possible to determine (estimate) the Strain (ε) value through the angular coefficient of the line for each sample. The values obtained for the Strain ( ε) as well as its associated error (σε) can be seen in Table 4.
As can be seen in Table 4, the strain value tends to decrease according to the decreasing ionic radius, that is, ε = 0.078, 0.069, and 0.062 for the LaCrO 3 , GdCrO 3 and YCrO 3 structures, respectively. This result is related and in agreement with the volume of the unit cell, where we observe (see Table 1) that the volume decreases according to the decrease in the ionic radius, that is, the tensions of the crystalline unit cell tend to decrease due to the decrease of the lattice parameters so the strain also tends to decrease. This suggests, for example, that the YCrO 3 compound can accommodate doping or substitutions ions more easily in its structure, while LaCrO 3 (with a higher strain) is more subject to a structural phase transition in the same circumstance.

Conclusion
In summary, was performed a comparative study of the effect of ionic radius on the structural properties of orthochromites RCrO 3 (R = La, Gd, and Y). It is observed that the b and c lattice parameters increase, whereas a lattice parameter decreased, as the ionic radius of R 3+ increased, and by consequence, an increase in the unit-cell volume is observed. By analyzing the angles and bond lengths, the same increase with an increase in ionic radius, the inclination and rotation angles tend to decrease. The distortion parameter suffers fluctuation in its value according to the increased ionic radius. The strain for the orthochromites decreased according to the decrease in the ionic radius. Thence, the main objective of this work, which was to describe in detail how to obtain important parameters in the structural analysis of materials with perovskite structure, in order to help research beginners in this area, was achieved successfully.