Empirical Formula of Lattice Constant and Tolerance Factors of A2BSbO7 (A3+ = Y, Dy, Gd, Bi; B3+ = Fe, Ga) Pyrochlore Solid Solution

In this work, we apply the concept of tolerance factors (t) to pyrochlore solid solution, particularly when the B-site contains two different ions with different masses (as well as charge states) in the formula unit \({A}_{2}^{3+}{B}^{3+}{B}^{{{\prime}}5+}{O}_{7}\) as in the present sample of A₂BSbO₇ (A³⁺ = Y, Dy, Gd, Bi; B³⁺ = Fe, Ga). We examine the previous tolerance factor and proposed a model of lattice constant (a) that depends only on the ionic radii R_A, R_B (= \(\frac{R\left(Fe\right)+R(Sb)}{2}\) or \(\frac{R\left(Ga\right)+R(Sb)}{2}\)) and R_O. Then we proposed an empirical tolerance factor (t), that depends only on the \({R}_{A},{R}_{B}\,{\mathrm{ and }\,R}_{O}\) of the constituent atoms. We discuss the structural stability field and property features of mixed pyrochlore oxide compounds before measuring their structural data as for the case of perovskites.

Graphical abstract

In the present work, we have proposed the empirical formula of lattice constant of A₂BSbO₇ (A³⁺ = Y, Dy, Gd, Bi; B³⁺ = Fe, Ga) of formula unit \({A}_{2}^{3+}{B}^{3+}{B}^{{{\prime}}5+}{O}_{7}\) and hence find the tolerance factor, which predict the structural stability field and property features of mixed pyrochlore oxide compounds before measuring their structural data as for the case of perovskites. Caption: Errors (in %) between the calculated and experimental lattice constants. In the above figure, we shown that the errors (%) between the calculated and experimental lattice constants found from empirical lattice formula a = \(\frac{8}{\sqrt{3}}\left[1.4474357143\left({R}_{A}+{R}_{O}\right)-0.42931\frac{{\left({R}_{A}+{R}_{O}\right)}^{2}}{({R}_{B}+{R}_{O})}\right]\) and Rietveld analysis of powder diffraction data respectively. The error of predicting for the lattice constant by Brik and Srivastava (J Am Ceram Soc 95:1454–1460, 2012) is moderately higher than the error (≤ 0.523%) obtains by our model for mixed pyrochlore oxides.