Regular cubes

Figure 8 SEM picture of a single felodipine crystal (coarse grade). The regular cube shows an apparently smooth surface. The arrow indicates the point at which the next picture (Fig. 9) was taken. Source. From Ref. 10.

On the other hand, the crystalline field due to main symmetries other than Oh symmetry can be also related to this same case. For this purpose, it is useful to represent the octahedral structure of our reference ABe center as in Figure 5.5(a). In this representation, the B ions lie in the center of the six faces of a regular cube of side 2a and the ion A (not displayed in the figure) is in the cube center the distance A-B is equal to a. 

The advantage of this representation is that other typical arrangements can also be displayed using this regular cube, as shown in Figures 5.5(b) and 5.5(c). The... 

The regular cube used in Figure 5.5 to represent different symmetry centers suggests that these symmetries can be easily interrelated. In particular, following the same steps as in Appendix A2, it can be shown that the crystal field strengths, lODq, of the tetrahedral and cubic symmetries are related to that of the octahedral symmetry. Assuming the same distance A-B for all three symmetries, the relationships between the crystalline field strengths are as follows (Henderson and Imbusch, 1989) ... 

Thiourea forms MX (thiourea) 4 compounds if the lattice energy of MX is iess than 160 Kcal/mole, so that M = K, Rb, or Cs. Other large cations form similar compounds, for example, thallium and lead. The structures consist of columns of cations surrounded by eight sulphur atoms at the comers of an approximately regular cube. Each sulphur atom is shared by two cations. The S—C bonds lie in planes normal to the column, and the NHg groups project into channels which contain the anion, and sometimes water molecules (34). 

Thiourea complexes of the type MX(thiourea)4 have been found if the lattice energy of MX is less than 670 kJ mol-1. This leads to complexes of K+, Rb+ and Cs+, especially with Br and I- as anions. The polarizable thiourea separates the anions from the cations and each cation is probably surrounded by eight sulfur atoms at the corner of an approximately regular cube.65... 

Colourless crystals of KAg(SeCN)2 separated from aqueous solutions as regular cubes in addition to crystals of KjAg(SeCN)4 and finely acicular colourless crystals of K2Ag(SeCN)3. The latter salt was more readily prepared from alcoholic and acetone solutions. 

Figure 2.9 Relative energies of 3d orbital energy levels of a transition metal ion in low-symmetry distorted sites, (a) Regular octahedron (e.g., periclase) (b) trigonally distorted octahedron (e.g., corundum, spinel, approx, olivine M2 site) (c) tetragonally distorted octahedron (e.g., approx, olivine Ml site) (d) highly distorted six-coordinated sited (e.g., pyroxene M2 site) (e) regular cube (/) distorted cube (e.g., triangular dodecahedral site of garnet).

Equation 11.17 is the fundamental expression of the PARAFAC (parallel factor analysis) model [77], which is used to describe the decomposition of trilinear data sets. For nontrilinear systems, the core C is no longer a regular cube (ncr x ncc x net), and the non-null elements are spread out in different manners, depending on each particular data set. The variables ncr, ncc, and net represent the rank in the row-wise, columnwise, and tubewise augmented data matrices, respectively. Each element in the original data set can now be obtained as shown in Equation 11.18 ... 

Fig. 1 shows the TEM image of the investigated samples. The dark areas correspond to Co nanoparticles. Nanoparticles like regular cubes correspond to fee nanociystals. There are also hexagonal symmetry patterns that can be attributed to both hexagonal nanocrystals and to fee particles in the projection perpendicular to the spatial diagonal. Cobalt particles size distribution is shown in Fig. 2. Total particles number used to be analyzed the size distribution were about 700. The average size of cobalt nanoparticles is about 20 nm.

The operation of some of the other improper rotation axes can be illustrated with respect to the five Platonic solids, the regular tetrahedron, regular octahedron, regular icosahedron, regular cube and regular dodecahedron. These polyhedra have regular faces and vertices, and each has... 

Figure 4.5 The five Platonic solids (a) regular tetrahedron (b) regular octahedron (c) regular cube (d) regular icosahedron (e) regular dodecahedron. The point group symbol for each solid is given below each diagram...

The same procedure can be used to collect the symmetry elements for a regular octahedron, (Figures 4.5b, 4.7), described above, into the point group symbol 4/m 3 2/m. The regular cube, (Figure 4.5c), can be described with the exactly the same point group, 4/m 3 2/m. 

CUj is surrounded by eight oxygen atoms and situated in the center of regular cube,... 

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