title: Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake.
abstract: The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface fractals can be decomposed into a sum of surface mass fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, correlations can be built in the mass fractal amplitudes, which explains the decay of the scattering intensity I(q) ∼ qDs-4, with 1 < Ds< 2 being the fractal dimension of the perimeter. The curve I(q)q4-Dsis found to be log-periodic in the fractal region with a period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of the sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of the Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.
authors.affiliation: Joint Institute for Nuclear Research, Dubna 141980, Russian Federation. cherny@theor.jinr.ru and Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34051, Republic of Korea.
authors.forename: Alexander Yu
authors.identifier: http://orcid.org/0000-0001-6190-2815
authors.initials: AY
authors.lastname: Cherny
authors.affiliation: Joint Institute for Nuclear Research, Dubna 141980, Russian Federation. cherny@theor.jinr.ru and Horia Hulubei National Institute of Physics and Nuclear Engineering, RO-077125 Bucharest-Magurele, Romania.
authors.forename: Eugen M
authors.initials: EM
authors.lastname: Anitas
authors.affiliation: Joint Institute for Nuclear Research, Dubna 141980, Russian Federation. cherny@theor.jinr.ru.
authors.forename: Vladimir A
authors.initials: VA
authors.lastname: Osipov
authors.affiliation: Joint Institute for Nuclear Research, Dubna 141980, Russian Federation. cherny@theor.jinr.ru and Laboratory for Advanced Studies of Membrane Proteins, Moscow Institute of Physics and Technology, Dolgoprudniy, Russian Federation.
authors.forename: Alexander I
authors.initials: AI
authors.lastname: Kuklin