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The small-angle scattering (SAS) from the Cantor surface fractal on the planeand Koch snowflake is considered. We develop the construction algorithm for theKoch snowflake, which makes possible the recurrence relation for the scatteringamplitude. The surface fractals can be decomposed into a sum of surface massfractals for arbitrary fractal iteration, which enables various approximationsfor the scattering intensity. It is shown that for the Cantor fractal, one canneglect with a good accuracy the correlations between the mass fractalamplitudes, while for the Koch snowflake, these correlations are important. Itis shown that nevertheless, the correlations can be build in the mass fractalamplitudes, which explains the decay of the scattering intensity $I(q)\simq^{D_{\mathrm{s}}-4}$ with $1 < D_{\mathrm{s}} < 2$ being the fractal dimensionof the perimeter. The curve $I(q)q^{4-D_{\mathrm{s}}}$ is found to belog-periodic in the fractal region with the period equal to the scaling factorof the fractal. The log-periodicity arises from the self-similarity of sizes ofbasic structural units rather than from correlations between their distances. Arecurrence relation is obtained for the radius of gyration of Koch snowflake,which is solved in the limit of infinite iterations. The present analysisallows us to obtain additional information from SAS data, such as the edges ofthe fractal regions, the fractal iteration number and the scaling factor.

Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake