Ronin Institute for Independent Scholarship, c/o NSLS-II, Brookhaven National Laboratory, Bldg 745, PO Box 5000, Upton, NY, USA.
Ronin Institute for Independent Scholarship, Kirkland, WA, USA.
Acta Crystallogr A Found Adv. 2023 Jul 1;79(Pt 4):369-380. doi: 10.1107/S2053273323003121. Epub 2023 Jun 20.
Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on the three shortest non-coplanar lattice vectors) or by Delaunay-reduced cells (based on four non-coplanar vectors summing to zero and all meeting at obtuse or right angles) is commonly performed. The Niggli cell derives from Minkowski reduction. The Delaunay cell derives from Selling reduction. All are related to the Wigner-Seitz (or Dirichlet, or Voronoi) cell of the lattice, which consists of the points at least as close to a chosen lattice point as they are to any other lattice point. The three non-coplanar lattice vectors chosen are here called the Niggli-reduced cell edges. Starting from a Niggli-reduced cell, the Dirichlet cell is characterized by the planes determined by 13 lattice half-edges: the midpoints of the three Niggli cell edges, the six Niggli cell face-diagonals and the four body-diagonals, but seven of the lengths are sufficient: three edge lengths, the three shorter of each pair of face-diagonal lengths, and the shortest body-diagonal length. These seven are sufficient to recover the Niggli-reduced cell.
晶格的晶体学描述是结构解析、晶体学数据库搜索和串行晶体学中衍射图像聚类的重要工具。通常通过 Niggli 约化胞（基于三个最短的非共面晶格矢量）或 Delaunay 约化胞（基于四个非共面矢量，总和为零，且均在钝角或直角处相遇）对晶格进行描述。Niggli 胞源于 Minkowski 约化，Delaunay 胞源于 Selling 约化，它们都与晶格的 Wigner-Seitz（或 Dirichlet 或 Voronoi）胞有关，该胞由至少与所选晶格点一样接近任何其他晶格点的点组成。这里选择的三个非共面晶格矢量称为 Niggli 约化胞边。从 Niggli 约化胞开始，Dirichlet 胞由通过 13 个晶格半边缘确定的平面来描述：三个 Niggli 约化胞边的中点、六个 Niggli 约化胞面对角线和四个体对角线，但其中七个长度就足够了：三个边长度、每对面对角线长度中较短的三个，以及最短的体对角线长度。这七个长度足以恢复 Niggli 约化胞。
