Lattice parameter determination by using fitting in Hough space of CBED pattern

Department of Crystalline Materials Science, Nagoya University
â—‹yosifumi yasuda Koh Saito

The convergent-beam electron diffraction (CBED) method, which uses a nanometer-size electron probe, is a powerful tool for characterizing crystal structures in nanometer region. The method provides a diffraction pattern composed of reflection discs. The 000 disc shows dark lines, along which higher-order Laue zone (HOLZ) reflections satisfy the Bragg conditions. Since the positions of the dark lines (HOLZ lines) are sensitive to the variation of lattice parameters, lattice parameters can accurately be determined by comparing experimental HOLZ line positions with simulated ones. So far, distances between HOLZ line intersections have been used as a criterion of the fit of the experimental and simulated patterns [1,2]. However, it needs a cumbersome procedure to get intersections of many HOLZ lines and to sum up their distances.
In this paper, we propose a simple method to fit experimental and simulated HOLZ line positions using the Hough transform [2]. Since a HOLZ line is transformed into a spot by the Hough transform, the fit of HOLZ line positions in CBED patterns correspond to the fit of spot patterns in the Hough transforms. We define the error sum of squares (chi-sq.) as a sum of the distances between experimental and simulated Hough spots. Lattice parameters, acceleration voltage are determined by minimizing chi-sq. The kinematical approximation with an acceleration voltage correction is used for simulating HOLZ line positions to reduce a computing cost. The process of finding the best fit is automated by using Powell's method. We confirmed that the accuracy is a few 0.0001 nm from a [107] GaAs pattern, which is comparable to the intersection method.

[1] J. M. Zuo, Ultramicrosc., 41, 211 (1992).
[2] S. Kraemer and J. Mayer, J. Microsc., 194, 2 (1999).