An algebraic approximation, of order K, of a polyhedron correlation function (CF) can be obtained from γ′′(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 the left and right expansions of γ′′(r) around D_i−1 and D_i up to the terms O(r − D_i−1)^K/2 and O(D_i − r)^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.

中文翻译：

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基于弦长分布的多面体相关函数的代数近似

可以从γ'′（r），其弦长分布（CLD）首先考虑子区间[ D _{i -1}，  D _i中的]，得到多面体相关函数（CF）的K阶代数近似。]]>两个距离（r - D _{i -1}）^1/2和（D _i - r）^1/2的多项式的多项式，使得其围绕r = D _{i -1}和r = D展开_一世同时与D' _{- 1}和D _i周围的γ''（r）的左右扩展一致，直到项O（r - D _{i -1}）^{K / 2}和O（D _i - r）^{K / 2}， 分别。然后，对每个i，对多项式进行两次积分，并确定与最终端点处的所得积分匹配的积分常数。所得代数CF逼近的3D傅里叶变换可以正确重现q的精确形状因子的渐近行为，直到项O [ q- ^{（K / 2 + 4）} ]。为了说明起见，将过程应用于立方体，四面体和八面体。