Octahedral plane

The Group IV elements also show a linear correlation of their octahedral shear moduli, C44(lll) with chemical hardness density (Eg/2Vm).This modulus is for for shear strains on the (111) planes. It is a measure of the shear stiffnesses of the covalent bonds. The (111) planes lie normal to the bonds that connect the atoms in the diamond (or zinc blende) structure. In terms of the three standard moduli for cubic symmetry (Cn, Q2, and C44), the octahedral shear modulus is given by C44(lll) = 3CV1 + [4C44/(Cn - Ci2)]. Since the (111) planes have three-fold symmetry, they have only one shear modulus. The bonds across the octahedral planes have high resistance to shear which probably results from electron correlation in the bonds (Gilman, 2002). 

From the six equivalent points 100 there are thus produced six planes which enclose a cube of side 1 a and volume (1/tf)3. The density of the states is also 1 per (reciprocal) unit volume. This cube thus encloses (1/tf)3 states. A crystal with dimensions 1X1X1 cm with a simple cubic lattice contains (1 a)z atoms so that there is also 1 state available per atom in this zone, or there is space for 2 electrons per atom. If, for example, the reflection from the octahedral planes 111 is, as in... 

BCC real-space lattices are completely determined by the condition that each inner vector, k, go over into another by all the symmetry operations. This is not the case for the tmncated octahedron. The surface of the Wigner-Seitz cell is only fixed at the truncating planes, not the octahedral planes. Nonetheless, the volume enclosed by the truncated octahedron is taken to be the first BZ for the FCC real-space lattice (Bouckaert et ak, 1936). The special high-symmetry points are shown in Table 4.5. 

A similar conclusion has been reached for different reasons by Van Hardeveld and Van Montfoort (33). These authors consider what happens when additional atoms are added to the perfect octahedral planes of Poltorak s crystallites. In particular, they look for the appearance at the surface of special adsorption sites (B5) where an adsorbate... 

Certain important crystal planes are often referred to by name without any mention of their Miller indices. Thus, planes of the form 111 in the cubic system are often called octahedral planes, since these are the bounding planes of an octahedron. In the hexagonal system, the (0001) plane is called the basal plane, planes of the form lOTO are called prismatic planes, and planes of the form lOTl are called pyramidal planes. 

For y > 0.25, the increase of the vacancy concentration merely leads to a progressive addition of defect planes, i.e. to an increase in the number of tetrahedral layers with respect to the number of octahedral planes. 

The shear stress on the octahedral planes (the eight planes making equal angles with the principal planes) is found to be related to the principal stresses by (6)... 

Figure 4 gives an Arrhenius plot of the reaction rate for three HCOOH decomposition experiments with (111) and three with (100) discs. In all runs the decomposition was zero-order and the activation energy amounted to (23.6 0.5) kcal/mole. The interesting feature was that on the octahedral planes the reaction rate was three to four times higher yet, the number of sites per cm2 on the two types of planes does not differ very much (100) 1.44 x 1015 sites, (111) 1.78 x 1015 sites. In the decomposition of hydrazine, however, the opposite was observed (Fig. 5) ... 

The interesting polymorphism observed in sulfur is consistent with previous predictions. Bridgman (19) named sulfur as a possible candidate for new irreversibly created high pressure and high temperature modifications. Von Hippel (I) showed that the structures of selenium and tellurium are interrelated to the structure of polonium and can be developed by displacing the atoms of the octahedral planes or sliding the (111) planes in the polonium structure. He reported that at ambient conditions... 

Miroliubov [391] considers that the strength of a material is defined by its shearing resistance in the octahedral plane. The criterion is a quadratic function of the components of the spherical stress tensor. This function for (T2 = 0 gives... 

Fig. 3.59 The cross-slip of a screw dislocation a schematic constriction in an extended dislocation, making cross-slip possible b the stages of cross-slip from one octahedral plane to an other in (a) a SF before cross-slip (b) the partial dislocations of the extended dislocation combined along distance 1 (c) the dislocation, after combination of the partials curves under the influence of stress, cross-slipping into the ill cross-slip plane and (d) the extended screw dislocation and the SF continue on from one slip plane into another...

Development of the octahedral shear stress can be found in many texts and will not be given here. However, it is appropriate to note the geometry of the octahedral plane. That is, if a diagonal plane is identified for stressed element as shown in Fig. 2.19(a) such that the normal to the diagonal plane makes an angle of 54.7°, the stress state will be as shown in Fig. 2.19(b). The resultant shear stress on this octahedral plane, so named because there are eight such planes at a point, is the octahedral shear stress. 

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