A method is proposed of calculating the geometric term of the structure factor for quasicrystals, which enables incorporation of the phason disorder. The scheme is based on the series expansion of the structure factor with moments of the distribution function as coefficients. A distribution function is a mathematical object that is constructed for reference vertices of the tiles in the quasilattice. It encloses the entire structural information of the underlying quasilattice, together with the inherent disorder, necessary to calculate the diffraction pattern. By tuning the value of the distribution moments through the refinement procedure, it is possible to obtain a very good agreement of this new model of the decagonal AlCuRh phase with the experimental data, reflected in the crystallographic R factor of 6.08%. The characteristic bias of the calculated diffraction peak intensities observed for the low-intensity reflections is significantly diminished, confirming its origin being, to some extent, related to phason disorder. Additionally, it is no longer necessary to use the general Debye–Waller factor for phasons, as the new formula accommodates this type of structural disorder. However, the best result was obtained for the model combining the new approach with the Gaussian corrective term.

中文翻译：

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一种解决十边形准晶相子无序的新方法：AlCuRh 平铺分布函数的矩级数展开

提出了一种计算准晶结构因子几何项的方法，该方法能够纳入相子无序。该方案基于以分布函数的矩为系数的结构因子的级数展开。分布函数是为准晶格中瓦片的参考顶点构造的数学对象。它包含了底层准晶格的整个结构信息，以及计算衍射图案所必需的固有无序。通过细化过程调整分布矩的值，可以使这种新的十边形 AlCuRh 相模型与实验数据非常吻合，反映在晶体学 R 因子为 6.08%。对于低强度反射观察到的计算衍射峰强度的特征偏差显着减少，证实其起源在某种程度上与相子无序有关。此外，不再需要对相子使用一般的德拜-沃勒因子，因为新公式适应了这种类型的结构无序。然而，将新方法与高斯校正项相结合的模型获得了最好的结果。