Delaunay-Ito

Figure 5.13. The schematic of Delaunay-Ito transformation. Left - the four unit cell vectors (vj, V2, V3 and V4 = -V1-V2-V3) are associated with the comers of the tetrahedron, while the six scalar products between the corresponding pairs of the vectors (s through S34) are linked to the edges of the tetrahedron. Assuming that > 0, the transformation is carried out as shown on the right the sign of 12 is changed its value is subtracted from that on the opposite edge and added to those on all adjacent edges the direction of vi (or V2) is reversed and the new vectors v 3 and v 4 are determined as V3+V1 and V4+V1 (or V3+V2 and V4+V2, respectively).

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. 

Regardless of which indexing method was employed, the resulting unit cell (especially when it is triclinic) shall be reduced using either Delaunay-Ito or Niggli method in order to enable the comparison of different solutions and to facilitate database and literature searches. Furthermore, the relationships between reduced unit cell parameters must be used to properly determine the Bravais lattice. The Niggli-reduced cell is considered standard and therefore, is preferable. 

This algorithm realizes a zone search indexing method combined with the Delaunay-Ito technique (see section 5.10.1) for the reduction of the most probable unit cell. The most commonly used versions of computer codes are ITO 13 and IT015. The program arrives at a solution by using the following algorithm ... 

Reduces the resultant unit cell by using the Delaunay-Ito method and then transforms it into one of the 14 Bravais lattices if the lattice is not primitive. 

If no external evidence is available, it is still possible to determine the unit cell dimensions of crystals of low symmetry from powder diffraction patterns, provided that sharp patterns with high resolution are avail able. Hesse (1948) and Lipson (1949) have used numerical methods successfully for orthorhombic crystals. (Sec also Henry, Lipson, and Wooster, 1951 Bunn 1955.) Ito (1950) has devised a method which in principle will lead to a possible unit cell for a crystal of any symmetry. It may not be the true unit cell appropriate to the crystal symmetry, but when a possible cell satisfying all the diffraction peaks on a powder pattern lias been obtained by Ito s method, the true unit cell can be obtained by a reduction process first devised by Delaunay (1933). Ito applies the reduction process to the reciprocal lattice (see p. 185), but International Tables (1952) recommend that the procedure should be applied to the direct space lattice. 

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