平面上Cantor曲面分形与Koch雪花的小角散射
Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake
摘要：
研究了平面上Cantor曲面分形和Koch雪花的小角散射。我们发展了Koch雪花的构造算法，使散射振幅的递推关系成为可能。表面分形可以分解为表面质量分形的总和，用于任意分形迭代，这使得各种近似的散射强度。结果表明，对于康托分形，可以忽略质量分形振幅之间的相关性具有良好的精度，而科赫雪花，这些相关性是重要的。它示出，然而，相关性可以建立在质量分形振幅，这解释了衰减的散射强度$I(q)\sim q^{D_{\mathrm{s}}-4}$与$1 < D_{\mathrm{s}} < 2$是分形维数的周长。发现曲线$I(q)q^{4-D_{\mathrm{s}}}$在分形区域是对数周期的，其周期等于分形的标度因子。对数周期性来自于基本结构单元大小的自相似性，而不是它们之间的距离的相关性。得到了Koch雪花的回转半径的递推关系式，并在无限次迭代极限下求解。本分析允许我们从SAS数据中获得额外的信息，如分形区域的边缘，分形迭代次数和比例因子。
英文摘要：
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 a good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, the correlations can be build in the mass fractal amplitudes, which explains the decay of the scattering intensity $I(q)\sim q^{D_{\mathrm{s}}-4}$ with $1 < D_{\mathrm{s}} < 2$ being the fractal dimension of the perimeter. The curve $I(q)q^{4-D_{\mathrm{s}}}$ is found to be log-periodic in the fractal region with the period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of 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.
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