Abstract

There are several theoretical approaches for describing the cut-off effect on small-angle scattering curves of mass fractals because of the finite size of the scattering object (e.g., developed cluster of nanoparticles). The problem is to relate the power-law scattering, I(q) = Bq^–D (corresponding to the fractal dimension D), at high q-values with the Gunier approximation, I(q) = G exp(\( - R_{{\text{g}}}^{2}\)q²/3), at low q-values. Various empirical or semi-empirical functions are used to cut-off correlations in the direct space with the following Fourier transform into the q-dependence of the scattered intensity. Then, the selected approach is verified by comparing the calculated intensity and the model or experimental scattering data. However, up to now no well-proven criterion of correctness was formulated. Here, we employed the fact that the parameters of the two scattering levels are D-dependent, which can be used as an additional test for the correctness of the above approaches. The analysis of the universal combined parameter \({{BR_{{\text{g}}}^{D}} \mathord{\left/ {\vphantom {{BR_{{\text{g}}}^{D}} G}} \right. \kern-0em} G}\) as a function of D revealed different behavior for various models. The true dependence was obtained using the calculations for clusters generated by the algorithm, which allows regulating D. The simulated series of clusters with different sizes and packing densities gave the same predicted universal D-dependence of the combined parameter. The comparison showed that no theoretical data fit the simulations. One can see either principally different behavior, especially as D → 3, or some kind of renormalization is required. The given analysis is considered as a step towards understanding scattering by multiscale systems at a new level.

中文翻译：

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关于随机质量分形对小角度散射的尺寸截止效应

摘要

由于散射物体（例如，发达的纳米粒子簇）的大小有限，有几种理论方法可用于描述对质量分数维的小角度散射曲线的截止效应。问题在于，在高q值处的功率定律散射I（q）= Bq ^{– D}（对应于分形维数D）与Gunier近似值I（q）= G  exp（\（-R_ {{\ text {g}}} ^ {2} \）q ^2/3），低q值。各种经验或半经验函数用于在直接空间中截断相关性，随后将傅立叶变换转换为散射强度的q相关性。然后，通过比较计算的强度和模型或实验散射数据来验证所选方法。但是，到目前为止，尚未制定出行之有效的正确性标准。在这里，我们采用了两个散射级别的参数取决于D的事实，可以将其用作上述方法的正确性的附加测试。通用组合参数\（{{BR _ {{\ text {g}}} ^ {D}} \ mathord {\ left / {\ vphantom {{BR _ {{\ text {g}}}} ^ {D }} G}} \ right。\ kern-0em} G} \）作为D的函数揭示了各种模型的不同行为。使用通过该算法，这允许调节生成的群计算得到真相关d。不同大小和堆积密度的模拟簇系列给出了组合参数的相同预测通用D依赖性。比较表明没有理论数据适合模拟。可以看到基本上是不同的行为，尤其是当D →3时，或者需要某种重新规范化。给定的分析被认为是迈向理解多尺度系统散射的新台阶。