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Most industrial diamonds are distorted octahedral crystals bounded by eight faces with smaller faces around the edges as shown in Figure 220 (a) Figure 220 (b) (d) shows an ideal octahedron or a perfect diamond crystal and a simplified diagram of its hard and soft axes, lying exactly 90 deg to each other [25] The arrow in Figure 220 (b) points to directions of motion (of the workpiece against the tool) that will cause the least wear in the diamond Atomic or crystallographic planes are layers of atoms or planes along which atoms are arranged The relation of a set of planes to the axes of the unit cell is designated by Miller indices One corner of the unit cell is assumed to be the origin of the space coordinates, and any set of planes is identified by the reciprocals of its intersections with these coordinates The unit of the coordinates is the intersections with these coordinates and is the lattice parameter of the crystal If a plane is parallel to an axis, it is said to intersect it at infinity Figure 221 shows the cubic system in which the crosshatched plane BCHG intersects the Y-axis at one unit from the origin and is parallel to the X and Z-axes or intersects them at infinity Thus the data can be presented as in Table 23 [25] The illustrated plane has Miller indices of (010) If a plane cuts any axis on the negative side of the origin, then the index will be negative and is indicated by placing a minus sign above the index, as in (h k l) For example, the Miller indices of plane ADEF that goes through the origin (point A)
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Fig 220: An octahedral representation of a diamond crystal [25]
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52 Table 23
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X Intersection Reciprocal Miller indices : 1 0
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Y 1 1 1 1
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Z : 1 0
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Fig 221: Miller indices [26]
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cannot be determined without changing the location of the origin Any point in the cube may be selected without changing the location of the origin For convenience, let us take point B Plane ADEF is parallel to the X (BC) and the Z-axis (BG) but intersects the Y-axis at 1 The plane therefore has Miller indices of (0 1 0) As another illustration, the Miller indices of plane BDJ (Figure 221) may be determined as in Table 24 Table 24
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X Intersection Reciprocal Miller indices 1 1 1 1 Y 1 1 1 1 Z 1
This plane has Miller indices of (112) If the Miller indices of a plane result in fractions, then these fractions must be cleared For example, consider a plane that intersects the X-axis at 1, the Y-axis at 3 and the Z-axis at 1 Taking reciprocals gives indices of 1, 1/3, and 1 Multiplying throughout by 3 to clear these fractions results in Miller indices of (313) for the plane All parallel planes have the same indices Parentheses around Miller indices, as in (hkl ), signify a specific plane or a set of parallel planes Braces signify a family of planes of the same form (which are equivalent in the crystal), such as the cube faces of a cubic crystal: {100} = (100) + (010) + (001) + ( 1 00) + (0 1 0) + (00 1)
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Reciprocals are not used to determine the indices of a direction In order to arrive at a point in a given direction, it is necessary to start at the origin and move a distance u times the unit distance a along the X-axis, v times the unit distance b along the Y-axis, and W times the unit distance c along the Z-axis If u, v and w are the smallest integers to accomplish the desired motion, they are the indices of the direction and are enclosed in square brackets as in [uvw]; a group of similar directions are enclosed in angular brackets as in u v w For example, in Figure 221, to determine the direction AC, starting at the origin (point A), it is necessary to move one unit along the X-axis to point D and one unit in the direction of the Y-axis to reach point C The direction AC would have indices of [110] In a cubic crystal, a direction has the same indices as the plane to which it is perpendicular An approximate idea of the packing of the atoms on a particular plane may be obtained by visualizing a single unit cell of the bcc and fcc structures Considering the atoms as the lattice points, the number of atoms on a particular plane would be as given in Table 25 Table 25