脂肪分形的小角散射
Small-angle scattering from fat fractals
摘要：
一些实验小角散射（SAS）数据的特点是由一系列的幂律衰减与任意递减的散射指数值。为了描述这些数据，在这里，我们开发了一个新的理论模型的基础上三维脂肪分形（集分形结构，但非零体积），并显示如何可以提取结构信息的基础分形结构。本文分析计算了一种新引入的脂肪分形模型的单分散和多分散SAS强度（分形形状因子和结构因子），并研究了其在动量空间中的性质。该系统是一个三维确定性的质量分形建立在著名的康托分形的扩展。该模型使我们能够解释一系列的幂律衰减和分别，广义幂律衰减（叠加的最大值和最小值的幂律衰减）与任意减少的散射指数的范围从零到三。我们表明，在该模型中，本分析使我们能够获得所有的分形区域的动量空间中的边缘，分形迭代的数量和分形维数和标度因子在每个结构层次的分形。我们应用我们的模型来计算分形的回转半径的解析表达式。所获得的表征脂肪分形的量与缩放因子随迭代次数的变化相关。
英文摘要：
A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.
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