The pair correlation function, g(r), is a fundamental descriptor of the inner structure of fractal aggregates of monomers. It provides a natural tool for studying physical properties involving two-point interaction (e.g. optics of aggregates). Several domains of distances between pairs of monomers have been identified. The fractal domain (in which g(r) is a power-law) is generally dominant for large aggregates. We show here that the local behavior of g(r)—involving monomers tangent to a given monomer—is necessary for most of the quantitative applications, even if that local domain is not directly related to fractal morphology. We derive a simple generic pair correlation function for fractal aggregates, depending on three structural parameters only: the fractal dimension, d_f ; the prefactor, k_f ; the local mean coordination number, Zˉ . Unlike the fractal dimension, the prefactor and the mean coordination number are not universal since they depend on many parameters of the generation process. We discuss the impact of the definite shape of g(r) on the aggregate structure factor profile (a measurable quantity through small-angle x-ray scattering). Improvement from the new g(r) shape is also illustrated deriving an analytical expression of the geometric cross section, G, of aggregates for fractal dimension <2 . Accuracy is checked by comparing with numerical data from aggregates of fractal dimension df=1.9 . On the up-to-date analytical expression of G, the contributions of the short- and long-range behaviours of g(r) are well separated, and it is clear that the local behavior of the pair correlation function is required to obtain accurate values of the geometric cross section. Thus, in addition to the fractal structure of aggregates, the local structure of aggregates (to which the prefactor k_f and the g(r) divergence at short distances are related) also appears important to accurately describe their physical features.

中文翻译：

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分形聚集体局部结构的重要性

配对相关函数，G（r），是单体分形聚集体内部结构的基本描述符。它为研究涉及两点相互作用的物理特性（例如聚集体的光学）提供了一个自然的工具。已经确定了单体对之间距离的几个范围。分形域（其中G（r）是幂律）对于大总量通常占主导地位。我们在这里展示了本地行为G（r）——涉及与给定单体相切的单体——对于大多数定量应用来说是必要的，即使该局部域与分形形态不直接相关。我们导出了一个简单的分形聚合的通用配对相关函数，仅取决于三个结构参数：分形维数，df_​ ;前因子，kf_​ ;局部平均配位数， Zˉ 。与分形维数不同，前因子和平均配位数并不通用，因为它们取决于生成过程的许多参数。我们讨论确定形状的影响G（r）在聚集体结构因子分布上（通过小角度 X 射线散射可测量的量）。从新的改进G（r）还说明了形状，推导了几何横截面的解析表达式，G，分形维数的聚合体 <2 。通过与分形维数聚合的数值数据进行比较来检查准确性 dF=1.9 。最新的解析表达式G，短期和长期行为的贡献G（r）很好地分离，并且很明显，需要成对相关函数的局部行为才能获得几何横截面的准确值。因此，除了聚集体的分形结构之外，聚集体的局部结构（其前因子kf_​ 和G（r）短距离的散度相关）对于准确描述它们的物理特征也显得很重要。