The geometrical structure of grain boundaries in nanocrystalline material was analyzed in terms of power-law relationships. The power laws yielded exponents that were interpreted in terms of fractal-like dimensions. A box-counting fractal dimension was computed for 3 images that had been digitized from published transmission electron micrographs. Its average value was 1.70. An average site occupation probability was estimated, for the lattices in the images, by determining the relationship between the occupation probability and the fractal dimension for pseudo-random face-centered cubic lattices. The results of further numerical simulations suggested that the grain boundaries had a box-counting fractal dimension of 2.4.

Calculation of the Fractal Dimension of Grain Boundaries in Nanocrystalline Pd. J.Chadwick: Journal of Physics - Condensed Matter, 1999, 11[1], 129-33