Calculation of the fractal dimension of grain boundaries in nanocrystalline Pd

The geometric structure of grain boundaries in nanocrystalline Pd has been analysed in terms of power-law relationships. The power laws yielded exponents that were interpreted as fractal-like dimensions. The box-counting fractal dimension, (d) over bar(2d). was computed for three images digitized from published transmission electron micrographs; the average result was (d) over bar(2d) = 1.70 +/- 0.06. An average site occupation probability, p, was estimated for the lattices in the images, by determining the relationship between p and (d) over bar(2d) for pseudo-random fee lattices. The results of further numerical simulations suggested that the grain boundaries had a box-counting fractal dimension of (d) over bar(3d) = 2.4 +/- 0.3. The extent to which fractal theory is valid for nanocrystalline Pd is evaluated.