Sigma-Boundary Statistics by Length and Number

Abstract

This paper presents a rationale for comparative use of length fraction and number fraction statistics in grain boundary analysis from orientation maps generated by electron back-scatter diffraction (EBSD). The length and number fraction statistics for Σ3ⁿ coincidence site lattice (CSL) boundaries were measured and compared. The length fraction of Σ3 boundaries was 0.48 whereas the number fraction was significantly less, 0.36. A simple model was generated to estimate both the length fraction and number fraction of annealing twins (a subset of Σ3). The model showed that the number fraction of twins is 0.68, 0.75, 0.79 and 0.82 of the length fraction for 1, 2, 3 and 4 twins-per-grain respectively. For the experimental data the number fraction was 0.76 of the length fraction, implying that there were on average two twins-per-grain. In contrast to the Σ3 case, the length fraction for Σ9 and Σ27 boundaries was less than the number fraction. There are more inaccuracies involved in obtaining the number fraction than in obtaining the length fraction from EBSD maps, therefore the length fraction should be recommended as the standard reporting method. However a knowledge of the distribution in the microstructure of Σ3ⁿ segments is often crucial to the inquiry in addition to the length fraction.