Fractal-based description for the three-dimensional surface of materials

Abstract

An algorithm called variation-correlation analysis, used to estimate fractal dimension with good accuracy, has been developed. Applying this model to images of the atomic force microscope, magnetic force microscope, and scanning electron microscope, it has been demonstrated that there exists a fractal characteristic length εmax. When the scale ε is within εmax, the variation-correlation Vcor(ε) of the dimensionless field-like variable H(x,y), which may denote the height of a surface or the magnetic domain or the angle distribution, obey a power law, while when ε is over εmax, Vcor (ε) becomes constant for a given image. The concept of "fractal measure" MF is given, MF=(1-δ)/(1+δ), where δ is defined as the dispersed degree of points on a log-log plot. MF is a sort of linear measure of point distribution, which can be used to determine the fractal characteristic length. Investigation shows that the fractal dimension in the range ε