Читать книгу Materials for Biomedical Engineering - Mohamed N. Rahaman - Страница 107

Any direction in a crystal lattice (Figure 3.26) is described by:

 A line through the origin O of the unit cell that is parallel to the given direction;

 Resolving the length of this line along the three principal axes x, y, and z and expressing these lengths as a fraction of the unit cell dimensions a, b, and c;

 Converting the ratios to the three smallest integers, u, v, and w, for example

Figure 3.26 Diagram illustrating the specification of lattice directions in a crystal.

The convention used to describe the direction is to write the three integers u, v, and w in square brackets, that is, [uvw]. The direction LM in Figure 3.26, for example, is found by drawing the line OE through the origin that is parallel to it. Resolving the length of OE along the x, y, and z axes and expressing them as a fraction of the unit cell dimensions, we get

Thus, the direction LM is described as [111], as are directions parallel to LM or OE.

Similar to planes, we can have families of equivalent directions. In the cubic system, for example, the directions

are equivalent except for our arbitrary choice of axis labels and directions. They are denoted collectively as 〈111〉 directions where the integers are enclosed within angle brackets. In general, a family of equivalent directions described by the integers u, v, and w is written 〈uvw〉. Properties of a cubic crystal that depend on direction, such as the elastic modulus, for example, will be identical in these eight directions.