论文标题
多面体相关函数的代数近似于其和弦长度分布
Algebraic Approximations of a Polyhedron Correlation Function Stemming from its Chord Length Distribution
论文作者
论文摘要
可以从$γ\ pp（r）$（$γ\ pp（r）$，其和弦长度分布（CLD）中获得多面体相关函数（CF）的代数近似（CF），首先考虑在subinterval $ [d_ {i-1}，i-1}，i-1}，\，\，d_i] $ dectance of distance of distanmial int polynomial int py two factial int polynomial中$（r-d_ {i-1}）^{1/2} $和$（d_ {i} -r）^{1/2} $，以使其扩展在$ r = d_ {i-1} $和$ r = d_i $ y = d_i $ comcide and左侧，$γ$ d_ $ d_ $ d_ $ d_ $ d_ pp and $ d_ pp and $ d_i-1-1-1-1-1-1-1-1-1-1-1-1-1 $ d_ {i-d_; $ o \ big（r-d_ {i-1} \ big）^{k/2} $和$ o \ big（d_i-r \ big）^{k/2} $。然后，对于每个$ i $，一个集成了两倍的多项式，并确定在公共终点处与所得积分相匹配的集成常数。所得代数CF近似的3D傅立叶变换正确地复制了$ q $ s，确切形式的渐近行为至术语$ o（q^{ - （k/2+4）}）$。为了说明，该过程应用于立方体，四面体和八面体。
An algebraic approximation, of order $K$, of a polyhedron correlation function (CF) can be obtained from $γ\pp(r)$, its chord-length distribution (CLD), considering first, within the subinterval $[D_{i-1},\, D_i]$ of the full range of distances, a polynomial in the two variables $(r-D_{i-1})^{1/2}$ and $(D_{i}-r)^{1/2}$ such that its expansions around $r=D_{i-1}$ and $r=D_i$ simultaneously coincide with left and the right expansions of $γ\pp(r)$ around $D_{i-1}$ and $D_i$ up to the terms $O\big(r-D_{i-1}\big)^{K/2}$ and $O\big(D_i-r\big)^{K/2}$, respectively. Then, for each $i$, one integrates twice the polynomial and determines the integration constants matching the resulting integrals at the common end points. The 3D Fourier transform of the resulting algebraic CF approximation correctly reproduces, at large $q$s, the asymptotic behaviour of the exact form factor up to the term $O(q^{-(K/2+4)})$. For illustration, the procedure is applied to the cube, the tetrahedron and the octahedron.
