Calculation of the small-angle x-ray and neutron scattering from nonrandom (regular) fractals.

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The intensity of the small-angle x-ray or neutron scattering has been calculated for two nonrandom (regular) fractals: the Menger sponge and a related fractal, called the fractal jack (a form similar to the metal six-pointed object used in the American children's game). The scatterers are assumed to be systems of independently scattering, randomly oriented identical nonrandom fractals constructed from a material with uniform density. The scattered intensity I(q) can be expressed as a function of qa, where q=4\ensuremath{\pi}${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$sin(theta/2); \ensuremath{\lambda} is the scattered wavelength; theta is the scattering angle, and a is the edge of the cube which is the starting approximant to the fractal. The calculations show that I(q) is a monotonically decreasing function on which maxima and minima are superimposed. For large qa the monotonic decay is proportional to ${q}^{\mathrm{\ensuremath{-}}D}$, where D is the fractal dimension. The first maximum for q>0 is a single peak located at q=${q}_{1}$. Groups of maxima are found at q${=3}^{k}$${q}_{1}$, where k is a positive integer greater than 1. The number of maxima within a group becomes greater as k increases. Numerical calculations of I(q) provide no evidence that the maxima and minima are damped and die out as q becomes larger. Thus I(q) for the two nonrandom fractals does not appear to approach the simple power-law scattering proportional to ${q}^{\mathrm{\ensuremath{-}}D}$ which is characteristic of the small-angle scattering from random fractals. The techniques developed to calculate I(q) for the Menger sponge and the fractal jack can also be employed to find the small-angle scattering from other nonrandom (regular) fractals.