Small-angle scattering from porous solids with fractal geometry

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Abstract
The fractal or Hausdorff dimensionality of an ideal random structure exhibiting self-similarity implies that the intensity of radiation scattered at small scattering vectors should have a power-law dependence on the magnitude of the scattering vector. However, for any real system, the scattering law must be modified by the introduction of a correlation length which reflects the finite overall size of the system. This concept can also be incorporated into the outer (Porod) part of the small-angle scattering where a non-integral power-law dependence can arise from interfaces of fractal character. The scattering law for the inner region of the small-angle scattering (corresponding to larger distances) is proportional to Q-D where D is the fractal dimensionality of the structure. The scattering law for the outer region (corresponding to smaller distances) is proportional to QD-6, where in this case D is the fractal dimensionality of the interface. This reflects the crossover from scaling based on mass per unit volume to one based on mass per unit area as the length of the examining probe (the inverse scattering vector) becomes shorter. An inverse correlation is predicted between the fractal dimensionality of the interface and the correlation length obtained using a nonfractal (Debye) model over a restricted range of scattering vectors. Measurements on shaly (argillaceous) rocks exhibit deviations from Porod scattering indicative of microscopically rough interfaces with D>2, and show the negative correlation between interfacial dimensionality and correlation length.