Solid State PDF

SOLID STATE CRYSTALLOGRAPHIC DIRECTIONS A crystallographic direction is defined as a line between two points, or a vector. The following steps are used to determine the three directional indices: 1. A vector of convenient length is positioned such that it passes through the origin of the coordina...

SOLID STATE CRYSTALLOGRAPHIC DIRECTIONS A crystallographic direction is defined as a line between two points, or a vector. The following steps are used to determine the three directional indices: 1. A vector of convenient length is positioned such that it passes through the origin of the coordinate system. Any vector can be translated throughout the crystal lattice without alteration, if parallelism is maintained. 2. The lengths of the vector projections on each of the three axes are determined and measured in terms of the unit cell dimensions a, b, and c. CRYSTALLOGRAPHIC DIRECTIONS 3. These three values are multiplied or divided by a common factor to reduce them to the smallest integer values. 4. The three indices, written without commas, are enclosed in square brackets as [uvw]. The integers u, v, and w correspond to the reduced projections along the x, y, and z axes, respectively. CRYSTALLOGRAPHIC DIRECTIONS CRYSTALLOGRAPHIC DIRECTIONS CRYSTALLOGRAPHIC DIRECTIONS CRYSTALLOGRAPHIC DIRECTIONS CRYSTALLOGRAPHIC DIRECTIONS CRYSTALLOGRAPHIC DIRECTIONS CRYSTALLOGRAPHIC PLANES The orientations of planes for a crystal structure are represented in a similar manner. Again, the unit cell is the basis, with the three-axis coordinate system as represented in Figure 9. In all but the hexagonal crystal system, crystallographic planes Miller indices are specified by three Miller indices as (hkl).Any two planes parallel to each other are equivalent and have identical indices.The procedure employed in determination of the h, k, and l index numbers is as follows: 1.If the plane passes through the selected origin, either another parallel plane must be constructed within the unit cell by an appropriate translation, or a new origin must be established at the corner of another unit cell. CRYSTALLOGRAPHIC PLANES 2.At this point the crystallographic plane either intersects or parallels each of the three axes; the length of the planar intercept for each axis is determined in terms of the lattice parameters a, b, and c. 3.The reciprocals of these numbers are taken. A plane that parallels an axis may be considered to have an infinite intercept, and, therefore, a zero index. 4.If necessary, these three numbers are changed to the set of smallest integers by multiplication or division by a common factor. 5.Finally, the integer indices, not separated by commas, are enclosed within parentheses, thus: (hkl). CRYSTALLOGRAPHIC PLANES An intercept on the negative side of the origin is indicated by a bar or minus sign positioned over the appropriate index. Furthermore, reversing the directions of all indices specifies another plane parallel to, on the opposite side of and equidistant from, the origin. Several low-index planes are represented in Figure 12. One interesting and unique characteristic of cubic crystals is that planes and directions having the same indices are perpendicular to one another; however, for other crystal systems there are no simple geometrical relationships between planes and directions having the same indices. CRYSTALLOGRAPHIC PLANES CRYSTALLOGRAPHIC PLANES CRYSTALLOGRAPHIC PLANES CRYSTALLOGRAPHIC PLANES CRYSTALLOGRAPHIC PLANES Atomic Arrangements The atomic arrangement for a crystallographic plane, which is often of interest, depends on the crystal structure. The (110) atomic planes for FCC and BCC crystal structures are represented in Figures 13 and 14; reduced-sphere unit cells are also included. Note that the atomic packing is different for each case. The circles represent atoms lying in the crystallographic planes as would be obtained from a slice taken through the centers of the full-sized hard spheres. A “family” of planes contains all those planes that are crystallographically equivalent—that are,having the same atomic packing. Atomic Arrangements Atomic Arrangements Atomic Arrangements Crystalline and Non-crystalline Materials SINGLE CRYSTALS For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal. All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature, but they may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled. If the extremities of a single crystal are permitted to grow without any external constraint, the crystal will assume a regular geometric shape having flat faces, as with some of the gem stones; the shape is indicative of the crystal structure Crystalline and Non-crystalline Materials A photograph of a garnet single crystal is shown in Figure 15.Within the past few years, single crystals have become extremely important in many of our modern technologies, in particular electronic microcircuits, which employ single crystals of silicon and other semiconductors. Figure 15 Photograph of a garnet single crystal that was found in Tongbei, Fujian Province, China. Crystalline and Non-crystalline Materials POLYCRYSTALLINE MATERIALS Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline. Various stages in the solidification of a polycrystalline specimen are represented schematically in Figure 16. Initially, small crystals or nuclei form at various positions. These have random crystallographic orientations, as indicated by the square grids. The small grains grow by the successive addition from the surrounding liquid of atoms to the structure of each. Crystalline and Non-crystalline Materials The extremities of adjacent grains impinge on one another as the solidification process approaches completion. As indicated in Figure 16, the crystallographic orientation varies from grain to grain. Also, there exists some atomic mismatch within the region where two grains meet; this area, called a grain boundary. ANISOTROPY The physical properties of single crystals of some substances depend on the crystallographic direction in which measurements are taken. For example, the elastic modulus, the electrical conductivity, and the index of refraction may have different values in the and directions. This directionality of properties is termed anisotropy, and it is associated with the variance of atomic or ionic spacing with crystallographic direction. Substances in which measured properties are independent of the direction of measurement are isotropic. The extent and magnitude of anisotropic effects in crystalline materials are functions of the symmetry of the crystal structure; the degree of anisotropy increases with decreasing structural symmetry—triclinic structures normally are highly anisotropic. ANISOTROPY The modulus of elasticity values at , , and orientations for several materials are presented in Table 5. For many polycrystalline materials, the crystallographic orientations of the individual grains are totally random. Under these circumstances, even though each grain may be anisotropic, a specimen composed of the grain aggregate behaves isotropically. Also, the magnitude of a measured property represents some average of the directional values. Sometimes the grains in polycrystalline materials have a preferential crystallographic orientation, in which case the material is said to have a “texture.” The magnetic properties of some iron alloys used in transformer cores are anisotropic—that is, grains (or single crystals magnetize in a (100)-type direction easier than any other crystallographic direction. ANISOTROPY Energy losses in transformer cores are minimized by utilizing polycrystalline sheets of these alloys into which have been introduced a “magnetic texture”: most of the grains in each sheet have a (100)-type crystallographic direction that is aligned (or almost aligned) in the same direction, which direction is oriented parallel to the direction of the applied magnetic field.