Madison A E, Madison P A
Peter the Great St. Petersburg Polytechnic University, ul. Polytechnicheskaya 29, 195251 Saint-Petersburg, Russia.
St. Petersburg Electrotechnical University 'LETI', ul. Professora Popova 5, 197376 Saint-Petersburg, Russia.
Proc Math Phys Eng Sci. 2019 Jan;475(2221):20180667. doi: 10.1098/rspa.2018.0667. Epub 2019 Jan 9.
A multiple-cell approach is discussed as a possible alternative to the higher dimensional crystallography of icosahedral quasicrystals. It is based on the Socolar-Steinhardt tiling combined with the quasi-unit cell model. Quasi-unit cells fill the space without gaps and overlappings similar to those in periodic crystals. Similarly, the atoms can occupy general and special positions. The alloy stoichiometry and the packing density can be calculated through the relative tile frequencies, which in turn are determined as the components of the Perron-Frobenius eigenvector of the corresponding substitution matrix. The calculation of the diffraction pattern reduces to the Perron projection of a special matrix, the entries of which reflect the contribution of each type of quasi-unit cell to the coherent scattering.
讨论了一种多胞方法，作为二十面体准晶体高维晶体学的一种可能替代方案。它基于索科拉尔 - 斯坦哈特平铺与准单胞模型相结合。准单胞填充空间时没有间隙和重叠，类似于周期性晶体中的情况。同样，原子可以占据一般位置和特殊位置。合金化学计量比和堆积密度可以通过相对平铺频率来计算，而相对平铺频率又被确定为相应置换矩阵的佩龙 - 弗罗贝尼乌斯特征向量的分量。衍射图案的计算简化为一个特殊矩阵的佩龙投影，该矩阵的元素反映了每种类型的准单胞对相干散射的贡献。
