Delaunay-Ito reduction

From the four possible triplets of resulting vectors (vi, v, V3 to V2, V3, V4), the one that has shortest vectors is selected because the angles among these vectors are closest to 90°. The Delaunay-Ito reduced primitive cells can be classified into 24 types according to the relationships between unit cell vectors and their scalar products. They are easily converted into one of the 14 Bravais lattices. For example, if V = V2 and s u = s 23 = 0 (an = a23 = 90°), the standard unit cell is orthorhombic C-centered, with lattice vectors, v° calculated as follows  

The unit cell reduction using Delaunay-Ito method can be easily automated as is done in the ITO indexing computer code, which is discussed in section 5.11. The Delaunay-Ito reduced unit cell, however, may not be the one with the shortest possible vectors, although the latter is conventionally defined as a standard reduced unit cell. 

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