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## Concepts of Crystal Geometry

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**X-ray diffraction analysis shows that the atoms in a metal**• crystal are arranged in a regular, repeated three-dimensional • pattern. • The most elementary crystal structure is the simple cubic lattice • (Fig. 9-1). Figure 9-1 Simple cubic structure.**We now introduce atoms and molecules, or “repeatable**• structural units”. • The unit cell is the smallest repetitive unit that there are 14 • space lattices. • These lattices are based on the seven crystal structures. • The points shown in Figure 9-1 correspond to atoms or groups • of atoms. • The 14 Bravis lattices can represent the unit cells for all • crystals.**Figure 9-2 (a) The 14 Bravais space lattices (P = primitive**or simple; I = body-centered cubic; F = face-center cubic; C = base-centered cubic**Figure 9-3 a) Body-centered cubic structure; b)**face-centered cubic structure.**Figure 9-4 Hexagonal**close-packed structure Figure 9-5 Stacking of close-packed spheres.**Three mutually perpendicular axes are arbitrarily placed**through one of the corners of the cell. • Crystallographic planes and directions will be specified with respect to these axes in terms of Miller indices. • A crystallographic plane is specified in terms of the length of its intercepts on the three axes, measured from the origin of the coordinate axes. • To simplify the crystallographic formulas, the reciprocals of these intercepts are used. • They are reduced to a lowest common denominator to give the Miller indices (hkl) of the plane.**For example, the plane ABCD in Fig. 9-1 is parallel to the**• x and z axes and intersects the y axis at one interatomic • distance ao. Therefore, the indices of the plane are • , or (hkl)=(010). Figure 9-1 Simple cubic structure.**There are six crystallographically equivalents planes of**• the type (100). • Any one of which can have the indices (100), (010), • (001), depending upon the choice of • axes. • The notation {100} is used when they are to be considered • as a group,or family of planes.**Figure 9.6(a)shows another plane and its intercepts.**Figure 9-6(a) Indexing of planes by Miller indices rules in the cubic unit cell**As usual, we take the inverse of the intercepts and multiply**them by their common denominator so that we end up with integers. In Figure 9.6(a), we have**Figure 9.6(b) shows an indeterminate situation. Thus, we**have to translate the plane to the next cell, or else translate the origin. Figure 9-6(b) Another example of indexing of planes by Miller rules in the cubic unit cell.