Unit cells reduced

Engelhard Corporation, Reduced Unit Cell Size Catalysts Offer Improved Octane for FCC Gasoline, The Catalvst Report, Tl-762. 

Table 9.1. The reduced unit cell volume, Viz, shrinks slightly with the ionic size of the transition metal as represented by the filled circles in Figure 9.8 where V is a unit cell volume and z is a z-number. A similar trend can be observed in the Sr series.

Table 9.1 Lattice parameters (nm) and reduced unit cell volumes (nm3) of A3MN3-type compounds...

Figure 9.8 Reduced unit cell volumes of A3MN3 (A = Ca, Sr, Ba M = V, Cr, Fe, Co) - type compounds vs ionic radius of M3+. Triangles, filled and open circles represent the unit volumes of Ba3FeN3, Ca3MN3 and Sr3MN3, respectively.

Figure 1 Some examples of fragments of strips (a) Square lattice strip or ladder of w=3, (b) polyphenantrene, (c) polyaceacene, (d) poly(benz[m,n])anthracene. The region between two dashed vertical lines define a (reduced) unit cell...

The strips may be divided into unit cells or eventually reduced unit cells (see Fig. 1), when the space group of the strip contains operations involving glide-reflections or screw-rotations, i.e. a combination of an improper reflection or two-fold rotation followed by a non-primitive translation of half a unit cell, which by themselves do not leave the lattice invariant. Each cell will contain w sites, w x L = 2N. 

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. 

The Niggli approach defines the reduced unit cell in terms of the shortest possible vectors. In other words ... 

Unfortunately, this simple condition cannot be evaluated directly, and in order to find a properly reduced unit cell it should be replaced by a total of five simultaneous inequalities ... 

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. 

Figure 18. Typical reduced unit cell enclosed within dotted lines also shown are the directions taken to make the C and r. counts of the text.

In contrast to the hexagonal-shaped P j -phase, another new phase, called the -phase, is also found in the products of diffusion synthesis [7]. The X i-phase is a rectangular plate-shaped crystal. The determination of the detailed structure is still in progress. Preliminary results show that the lattice parameters of the X -phase are similar to that of the 0-phase [3,4] with Z = 4. However, the new phase is characterized by its peculiar transport behavior (see below). Table 1 shows the structure parameters of the P - and phases, in comparison with the other isomers. It can be seen that the reduced unit cell volume in the X j-phase is a little larger than that of the 0-phase, indicating either the existence of excess iodine atoms, or the misarrangement of the iodine chain. In the following, we will present experimental results for the two new phases. 

Fig.5.2-14a-d Symmetry-reduced unit cell for Si(lll) 7x7 (DAS model). In (a), the unit cell (two triangular halves, one faulted (hatched) and one unfaulted) is reduced to smaller subunits and recomposed into a c-hexagonal cell. The atoms are drawn and numbered in (b) - (d). The numbering is the same as in Table 5.2-7 the origin of the coordinates is at atom 103. (b) The four adatoms (c) the first- and second-layer atoms (d) the third- and fourth-layer atoms [2.24,25]... 

Table 5.2-7 Coordinates of the adatoms and four uppermost layer atoms in the symmetry-reduced unit cell of Si(lll) 7x7 (see Fig. 5.2-14 for the numbering of the atoms). The origin is taken at atom 103. X-ray data give a value of 2.11A for the adatom-back atom projected bond length (between atoms 1 and 13 and between atoms 1 and 28 in the table) and a value of 2.49 A for the dimer bond length (betweens atoms 26 and 27 in the table) [2.24-26]...

The structure of Fortisan (cellulose - II) has been re-determined and re-analysed as a reduced cell, which is monoclinic with a = 0.446, b = 1.034 (fibre axis), c — 0.725 nm, and p — 98.6°. The chains are arranged parallel to one another and are all identical with one per unit cell. Apart from the reduced unit cell the structure is as determined previously. 

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