3D Microstructures of Sb₂Te₃ Precipitates in PbTe Matrix with Prediction by a Weak Compatibility Condition

Abstract

We propose that a weak compatibility condition predicts the elongated directions for Widmanstätten type precipitates. The distribution of the elongated directions of precipitates lies on a family of crystallographically equivalent cones in 3D determined by a certain transformation stretch matrix obtained independently. A 3D visualization and digitization method is developed to show how the cone variants control the preferred growth directions during precipitation of Sb₂Te₃ in a (5 µm)³ PbTe matrix. A series of two-dimensional secondary electron images are acquired along the direction perpendicular to the imaging plane. By pixelating all the images and calculating the position vectors on the surface of each precipitate, the elongation directions are calculated using a 3-dimensional ellipsoidal fitting for 182 precipitates. The 3D plot of the elongation directions shows that their spacial orientations are close to four predicted cones with a standard deviation of 5.6°. The length along the elongation directions reveals an asymmetric distribution with a mean value of about 240 nm. The total volume fraction of the precipitates is 8.3 %. The average area of the precipitates per volume is 0.68 µ^-1 by one point statistical calculation. These results build on our study presented in [1] by analyzing a significantly bigger data set and by including the length distribution and 1-point statistics.