A method is developed for analyzing the outer part of the small-angle x-ray or neutron

Small-angle X-ray and neutron scattering are important techniques for studying the structure of fractals and other disordered systems on a scale of lengths from about 10 to 2000 A. This review begins with a brief outline of some properties of fractals. The small-angle scattering from fractal systems is then discussed and the effect of polydispersity is considered. The intensity of small-angle scattering from fractals and other disordered systems is often proportional to a negative power of the quantity q = 4~,t-Isin(0/2), where 0 is the scattering angle and ,~ is the X-ray or neutron wavelength. From the magnitude of the exponent that describes this type of scattering, which is often called power-law scattering, much important information can be obtained. Some situations in which power-law scattering can be expected are described. To illustrate the scattering from fractals and disordered systems, several experimental investigations of mass-fractal silicas and porous solids are reviewed and some calculations of the small-angle scattering from model fractal systems are outlined.

The intensity of the small-angle X-ray and neutron scattering from a polydisperse system of randomly oriented independently scattering particles is shown to be proportional to h -" for all values of the scattering vector h when the distribution of particle dimensions is proportional to r -t2d+ 1 -~), where h =4rc2 -1 sin(0/2); 0 is the scattering angle; 2 is the wavelength; r is the maximum dimension of a particle; and d is the number of dimensions of the particles. The value of 0¢ lies in the interval 0<~

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The small-angle x-ray scattering from fully and partially derivatized porous silicas has been studied. Power-law-scattering exponents of magnitude greater than 4 have been found in all cases. The magnitudes of the exponents increased with the alkyl chain length and with the degree of surface derivatization. In a preliminary model to explain these observations, a power-law-scattering exponent with magnitude greater than 4 is related to a ‘‘fuzzy’’ pore boundary, in which the density varies continuously at the pore boundary instead of changing discontinuously from a value of zero in the empty pore to the essentially constant density characteristic of the bulk silica, as is usually assumed in analyses of the small-angle scattering from porous silicas.