Determination of the Compositional Fluctuation Area in Ternary Perovskite Solid Solutions Using Least Square Method

Abstract

A method for an estimation of the compositional fluctuation area in ternary solid solutions was developed. An elliptic area of the compositional fluctuation which is defined by constants, A, B and φ, was assumed in a ternary diagram. The maximum and minimum lattice spacings in the assumed area were figured out for several lattice planes. Values of (Δd/d)_i^cal, fluctuations of the lattice spacings, were calculated from these values. Values of β, which is the net width of the X-ray diffraction peak of the sample, were measured and the values of β cosθ were plotted against sinθ. Theoretically, each plot is on a line whose section at sinθ=0 is fixed by the crystallite size of the sample and whose slope is Δd/d. The section was determined by the least square method using (Δd/d)_i^cal estimated from the assumed elliptic area. Then ∑e_i², a sum of squares of the difference between the calculated value of β cosθ and the observed one, was determined. The values of ∑e_i² were evaluated for various values of A, B and φ systematically. The values of A, B and φ which give the minimum value of ∑e_i² were finally determined. The area which is settled by the values of A, B and φ determined thus is the compositional fluctuation area. The compositional fluctuation areas such determined agreed with those determined by the method reported previously. By this method, the compositional fluctuation area can be determined without data of diffractions whose dependence of lattice spacing on composition is the same, while it could not be determined by the previous method.