Crystallographic Directions and Planes

The following steps must be followed in order to specify a crystallographic direction  

The vector that defines the crystallographic direction should be situated in such a way that it passes through the origin of the lattice coordinate system. 

The projections of this vector on each of the three axis is determined and measured in terms of the unit cell dimensions, a, b, c, obtaining three integer numbers, n2, n3. 

These three numbers, enclosed in square brackets and not separated with commas, [nvw], denote the crystallographic direction. 

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Crystallographic directions and planes are important in both the characteristics and the applications of semiconductor materials since different crystallographic planes can exhibit significantly different physical properties. For example, the surface density of atoms (atoms per square centimeter) can differ substantially on different crystal planes. A standardized notation (the so-called Miller indices) is used to define the crystallographic planes and directions normal to those planes. 

The two previous sections discussed the equivalency of nonparallel crystallographic directions and planes. Directional equivalency is related to linear density in the sense that, for a particular material, equivalent directions have identical linear densities. The corresponding parameter for crystallographic planes is planar density, and planes having the same planar density values are also equivalent. 

Transport properties. Ionic conduction and atomic diffusion are usually easier along certain crystallographic directions or planes. 

Following a short induction period, the decomposition of thalhum(I) azide proceeded [35,65] at a constant rate = 150 kJ mol" between 507 and 529 K) and the zero-order rate constants were directly proportional to surface area. These kinetic observations were interpreted through reference to electron micrographs. Reaction occurred on preferred crystal planes, with the interface advancing predominantly in the (110) crystallographic direction and holes of uniform depth and constant cross-section were observed. 

In this paper, we present some of our results on Si nanowires and quantum dots. The PL lifetime is calculated as a function of the crystallographic direction and the size of the wires and the dots. We have considered sizes in the range of 1-4 nm and crystallographic directions varying in (100) plane from [100] to [110] direction. Here, we discuss representative results on wires and dots of size 2 nm. 

The body-centred cubic crystal is not close-packed. The slip systems with the closest packed directions and planes in this lattice are of the type 110 (111) (figure 6.15). With two slip directions per plane and six different slip planes, twelve slip systems result. As summarised in table 6.2, slip is also possible on other crystallographic planes that are only slightly more difficult to activate [55]. 

In many cases, it is necessary to specify directions and planes in a crystal lattice. It is most sensible to do so using a crystallographic coordinate system, with axes parallel to the edges of the chosen unit cell. All parallel directions and planes in a crystal are equivalent, rendering it unnecessary to state the origin of the direction vector or plane. 

For specifying directions and planes in a crystal, the origin of the crystallographic coordinate system is positioned in a lattice point, and the axes are scaled so that the length of every edge of the unit cell is one. Thus, for non-cubic lattice types, this coordinate system is non-Cartesian. 

Slip Plastic deformation by irreversible shear displacement of one part of crystal with respect to another in a definite crystallographic direction and on a specific crystallographic plane. 

Chapter 2 was concerned primarily with the various types of atomic bonding, which are determined by the electron structures of the individual atoms. The present discussion is devoted to the next level of the structure of materials, specifically, to some of the arrangements that may be assumed by atoms in the solid state. Within this framework, concepts of crystallinity and noncrystallinity are introduced. For crystalline solids, the notion of crystal structure is presented, specified in terms of a unit cell. The three common crystal structures found in metals are then detailed, along with the scheme by which crystallographic points, directions, and planes are expressed. Single crystals, polycrystalline materials, and noncrystalline materials are considered. Another section of this chapter briefly describes how crystal structures are determined experimentally using x-ray diffraction techniques. 

This concludes our discussion on crystallographic points, directions, and planes. A review and summary of these topics is found in Table 3.3. 

Point Coordinates Crystallographic points, directions, and planes are specified in terms of indexing schemes. The basis for the determination of each index is a coordinate axis system defined by the unit cell for the particular crystal structure. 

Plastic deformation of metals and alloys occurs by slip and/or twinning on certain crystallographic planes and along certain crystallographic directions and is accomplished by the movement of dislocations (Volume 2, Chapter 7). Plastic deformation results in considerable disorder, leading to phase transformations in some cases. The de-... 

The selective oxidation of ra-butane to give maleic anhydride (MA) catalyzed by vanadium phosphorus oxides is an important commercial process (99). MA is subsequently used in catalytic processes to make tetrahydrofurans and agricultural chemicals. The active phase in the selective butane oxidation catalyst is identified as vanadyl pyrophosphate, (V0)2P207, referred to as VPO. The three-dimensional structure of orthorhombic VPO, consisting of vanadyl octahedra and phosphate tetrahedra, is shown in Fig. 17, with a= 1.6594 nm, b = 0.776 nm, and c = 0.958 nm (100), with (010) as the active plane (99). Conventional crystallographic notations of round brackets (), and triangular point brackets (), are used to denote a crystal plane and crystallographic directions in the VPO structure, respectively. The latter refers to symmetrically equivalent directions present in a crystal. 

Fig. 2.8 Cleavage in the amphiboles. (A) Schematic representation of the characteristic stacked amphibole I-beams in the three-dimensional structure. A tetrahedral portion of an I-beam is labeled "silica ribbon." The octahedral portion is labeled "cation layer" and represented by solid circles. One of the possible cleavage directions (110) along planes of structural weakness is indicated by the line A-A stepped around the I-beams in the lower part of the diagram. (B) Cross section of the stacked I-beams with the directions of easy cleavage indicated. There is a lower density of bonds between I-beams in the crystallographic directions (110) and (110). These directions, parallel to the c axis and the length of the chains, are the planes of cleavage. The minimum thickness of a rhombic fragment produced through cleavage is 0.84 nm.

One SEXAFS specific feature is the polarisation dependence of the amplitude. This derives from the high anisotropy of the surface and of ultrathin interfaces, that we may consider as quasi two dimensional systems. The relative orientation of the X-ray electric vector with respect to the surface (interface) normal does represent a preferential excitation for those atom pairs aligned along the electric vector e.g. with the electric vector perpendicular to the surface (interface) plane the EXAFS amplitude will be maximum for the atom pairs aligned normal, or almost normal to the surface (interface). The electric vector can be also aligned, within the surface plane, along different crystallographic directions. 

Beside dislocation density, dislocation orientation is the primary factor in determining the critical shear stress required for plastic deformation. Dislocations do not move with the same degree of ease in all crystallographic directions or in all crystallographic planes. There is usually a preferred direction for slip dislocation movement. The combination of slip direction and slip plane is called the slip system, and it depends on the crystal structure of the metal. The slip plane is usually that plane having the most dense atomic packing (cf. Section 1.1.1.2). In face-centered cubic structures, this plane is the (111) plane, and the slip direction is the [110] direction. Each slip plane may contain more than one possible slip direction, so several slip systems may exist for a particular crystal structure. Eor FCC, there are a total of 12 possible slip systems four different (111) planes and three independent [110] directions for each plane. The... 

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