Crystal planes and directions

Crystals growing on a substrate may be oriented every which way that is, the direction axes of individual crystallites can be randomly distributed. However, where one particular axis is oriented or fixed in nearly one direction, we speak about a single texture. When two axes are thus fixed or oriented, we speak about double texture. Monocrystalline orientation refers to a scenario in which there are three such nearly oriented axes, including epitaxial films. Orientation here is viewed with respect to any fixed (in space) frame of reference. Crystal planes and directions are illustrated in Figure 16.5. A brief discussion of the enumeration of these elements follows. 

Figure 16.5. Crystal planes and directions as indicated. One plane and one direction in each cube are indicated. (From Ref 1, with permission from Noyes.)...

Goethite and lepidocrocite have recently been moved from Pbnm to Pnma and from Cmcm to Bbmm, respectively. As a result of this change, the crystal planes and directions in this book are different from those in the 1 edition. 

Fig.1.3 -I Some crystal planes and directions in a cubic crystal, and their Miller indices...

Figure 4.2 Shows the Miller indexing scheme for crystal planes and directions in a simple cubic crystal. Similar indexing methods are used in other Bravais lattices, although the basis vectors are not necessarily parallel to Cartesian coordinate axes. In hexagonal lattices an alternate labeling scheme employing four in ws in which only three of the four are independent is often used. Notice that the two (111) planes marked are adjacent and parallel to one another.

An account of the use of Miller indices to describe crystal planes and lattice directions is beyond the sco[>e of this article a very adequate treatment of this topic is, however, given in Reference 1. 

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. 

Crystal Locations, Planes, and Directions. In order to calculate such important quantities as cell volumes and densities, we need to be able to specify locations and directions within the crystal. Cell coordinates specify a position in the lattice and are indicated by the variables u, v, w, separated by commas with no brackets ... 

The Miller-Bravais index system for identifying planes and directions in hexagonal crystals is similar to the Miller index system except that it uses four axes rather than three. The advantage of the four-index system is that the symmetry is more apparent. Three of the axes, ai, a2, and a3, he in the hexagonal (basal) plane at 120° to one another and the fourth or c-axis is perpendicular to then, as shown in Figure 3.1. 

Stereographic projection provides a convenient way of displaying the angular relations between planes and directions in a crystal in two dimensions. The system involves first projecting planes and directions of interest onto a spherical surface and then mapping the spherical surface. Figure 4.1 illustrates how planes and directions are projected onto a sphere. If an infinitesimal crystal were placed at the center of a sphere and its planes extended, they would intersect the sphere as great circles and their directions would intersect the sphere as points. 

Mapping of planes and directions by placing an infinitesimal crystal at the center of a sphere and projecting planes onto the sphere to form great circles and lines to form points. 

Slip occurs along specific crystal planes (slip planes) and in specific directions (slip directions) within a crystal structure. Slip planes are usually the closest-packed planes, and slip directions are the closest-packed directions. Both face-centered-cubic (FCC) and hexagonal-close packed (HCP) structure are close packed structures, and slip always occurs in a close packed direction on a closepacked plane. The body-centered-cubic (BCC) structure is not, however, close packed. In a BCC system, slip may occur on several nearly close packed planes or directions. Slip planes and directions, as well as the number of independent slip systems (the product of the numbers of independent planes and directions), for these three structures are listed in Table 7.2. 

In some structures, several planes and directions may be equivalent by symmetry. For example, this is the case for the (100), (010), (001), (100), (010), and (OOl) planes in the diamond cubic structure. Equivalent directions are denoted concisely as a group by using angular brackets. Thus, the (100) directions in a diamond cubic lattice include all of the directions that are perpendicular to the six planes noted above. The Miller index notation thus provides a concise designation for describing the surfaces of semiconductor crystals. 

Stacking crystal planes and uniaxial growth directions... 

Provided that there is no additional surface charge, fj, is a pure bulk term which is independent of any electrostatic potential. The term is the contribution of surface dipoles [1, 2] (Fig. 2.1). Such a dipole can be caused by an unsymmetrical distribution of charges at the surface because there is a certain probability for the electrons to be located outside the surface. In the case of compound semiconductors, dipoles based on the surface structure caused by a particular ionic charge distribution occur. These effects depend on the crystal plane and on the reconstruction of the surface atoms [3, 4]. These dipole effects also influence the electron affinity and ionization energy. In the case of metals, the work function is a directly measurable quantity, and for semiconductors it is calculable from ionization measurements. However, the relative contributions of fi and ex are not accessible experimentally and data given in the literature are based on theoretical calculations (see e.g. ref. [1]). 

Planes, Directions and Plastic Deformation. When dealing with the modern interpretation of the mechanism of plastic deformation of metals and alloys and with the associated problems of hardness and strength—resistance to plastic deformation—we find it essential to be familiar with the more important crystallographic planes and crystallographic directions. We shall, therefore, with this in mind, devote our attention now to a brief discussion of the planes and directions of the atoms in the throe main types of metal crystal lattices... 

NiO grows on MgO with both grains aligned so that corresponding planes and directions in the two crystals are nearly parallel. The difference in lattice parameter (Aa) will be accommodated by a square array of misfit dislocations. 

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