Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 26

Directions in crystals and lattices are labelled by first drawing a line that passes through the origin and parallel to the direction in question. Let the line pass through a point with general fractional coordinates x, y, z; the line also passes through 2x, 2y, 2z; 3x, 3y, 3z, etc. These coordinates, written in square brackets, [x, y, z], are the indices of the direction; x, y and z are arranged to be the set of smallest possible integers, by division or multiplication throughout by a common factor. Thus [½½0], [110], [330] all describe the same direction, but by convention [110] is used.

For cubic systems, an [hkl] direction is always perpendicular to the (hkl) plane of the same indices, but this is only sometimes true in non‐cubic systems. Sets of directions which, by symmetry, are equivalent, e.g. cubic [100], [010], are written using angle brackets, <100>. Some examples of directions and their indices are shown in Fig. 1.15(e). Note that the [210] direction is defined by taking the origin at the bottom‐left‐front corner of the unit cell and taking the fractional coordinates: 1, 0.5, 0, to define the direction. For the direction [], where the bar sign indicates a negative direction, a different origin is chosen, at the bottom‐right‐front corner of the unit cell and passing through the fractional coordinates: –1, 0.67, 1.