Quantitative symmetry determination and symmetry mapping using convergent beam electron diffraction technique

Abstract

The symmetry recorded in convergent beam electron diffraction (CBED) patterns is in general determined by direct visual inspection [1], which does not provide uniform measurement. Furthermore, experimental CBED patterns are often noisy and deviate from the ideal symmetry because of the sample geometry and defects. Thus, the imperfection in experimental CBED patterns can lead to uncertainty in the symmetry determination [2]. Here, we propose a symmetry quantification method for CBED patterns using the profile R-factor (Rp) [3, 4] and the normalized cross-correlation coefficient (γ) [5]. We have also developed computer algorithms to automate these procedures. We demonstrate that the method proposed here is highly effective and provides a more precise way to determine the symmetry in CBED patterns. The symmetry quantification method can be also combined with a scanning electron diffraction technique for symmetry mapping [6]. Figure 1 shows the image processing procedures for mirror symmetry quantification. First, the symmetry related two diffraction discs (A, A’) are selected about the mirror plane (yellow line) as shown in Fig. 1(a). The template A is used as the reference motif so that the symmetry element is calculated by comparing with template A’. The template A and A’ are first aligned (Fig. 1(c)), and A’ is flipped horizontally to obtain a mirror image (A’m, Fig. 1(h)). For the rotational operation, the template A’ is simply rotated by 360/no with respect to n-fold rotation. The circular mask (Figs. 1(d) and (i)) is finally used to remove areas affected by CBED disk edge and to obtain the final templates (Figs. 1(e) and (j)). Then, we applied the profile R-factor (Rp) and the normalized cross-correlation coefficient (γ) to quantify the similarity between A (=IA(x, y)) and A’m (=IB(x, y)) as given in Eqs. (1) and (2).