## Surface fractal If the $q$ vector goes really big, we will have our signal representing even smaller structure in the particle. For this case, we could typically read "surface fractal" from the intensity curve. Also, due to the probing scale is already smaller than the particle size, we can hardly interpretate the result into structure factor and form factor, as stated in [[Structure factor S(q) and density autocorrelation functions#Understanding structure factor in different q ranges]]. Identify the range of $q$. Here re present the slope and the corresponding structures that is possible. The surface fractal follows: $S(q)\propto q^{-d}$ This $d$ is surface fractal. - $d=4$, smooth, randomly oriented surfaces. - $d$ between $3-4$, rough surfaces, surface fractal. About are discrete, colloidal-like structure. If we have fractals or similar structures (not discrete), we may have - $d\approx2$, planar objects, separating surface. - $d\approx 1$, elongated objects, cylinder, rod. It could be challenging to determine which part gives the proper slope. >[!Note] >Although not discussed here, but with surface fractal and Porod law, it is possible to determine the aspect ratio (or say surface/volume ratio) of microstructures. Check slides or related books for more information.