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CRYSTALLOGRAPHIC DIRECT IONS & PLANES

?CRYSTALLOGRAPHIC DIRECTIONS :

     A crystallographic direction is defined as a line between two points, or a vector. The following steps are utilized in the determination of 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 may be      translated throughout the crystal lattice without alteration, if           parallelism is maintained.

2. The length of the vector projection on each of the three axes is        determined; these are measured in terms of the unit cell                   dimensions a, b, and c.

3. These three numbers are multiplied or divided by a common            factor to reduce them to the smallest integer values.

4. The three indices, not separated by commas, are enclosed in           square brackets, thus: [uvw]. The u, v, and w integers correspond     to the reduced projections along the x, y, and z axes, espectively.

   For each of the three axes, there will exist both positive and negative coordinates.Thus negative indices are also possible, which are represented by a bar over the appropriate index

EXAMPLE 3-4:

   Determine the indices for the direction shown in the accompanying figure.

SOLUTION :

                                                           x                y                      z

Projections                                         a/2             b                    ? c

Projections (in terms of a, b,  c)          1/2             1                      0

Reduction                                          1                2                      0

Enclosure                                                         [120]

Notes:

  1- Equivalent directions are grouped together into a family, which       are enclosed in angle brackets, thus:‹100›.

  2- For some crystal structures, several nonparallel directions with         different indices are actually equivalent; this means that the            spacing of atoms along each direction is the same. For                     example, in cubic crystals, all the directions represented by the        following indices are equivalent: [100], [ 00], [010], [0 0],            [001], and [00 ].

  3- Directions in cubic crystals having the same indices without          regard to order or sign, for example, [123] and [ 1 ], are                equivalent. This is, in general, not true for other crystal systems.      For example, for crystals of tetragonal symmetry, [100] and            [010] directions are equivalent,  whereas [100] and [001] are          not.