Читать книгу Materials for Biomedical Engineering - Mohamed N. Rahaman - Страница 106

Any plane A′B′C′ can be defined by the intercepts OA′, OB′, and OC′ with the three principal axes of the crystal system (Figure 3.25). In the common notation, referred to as Miller indices,

 We take the reciprocals of the ratios of the intercepts to the corresponding unit cell dimensions

 Then the numbers are converted to the smallest integers

As an example, for a cubic crystal structure illustrated in Figure 3.25, the plane A′B′C′ has intercepts OA′ = 2a, OB′ = 3a, and OC′ = 3a. The reciprocals of the ratios of the intercepts to the corresponding cell dimensions are respectively

Converting these numbers to the smallest integers, we get 3, 2, and 2, respectively. These numbers are written in parentheses (322) and they are called the Miller indices of the plane. Thus, the plane A′B′C′ is referred to as the (322) plane. In general, a plane with Miller indices h, k, and l is written as (hkl).

A negative intercept of a plane on a crystal axis is written with a bar over the respective index. A plane that has intercepts of 2a, 3a, and −3a, respectively, on the x, y, and z axes, for example, has Miller indices of (32 ). A plane that is parallel to a principal axis of a crystal system is said to have an intercept of infinity (∞) on that axis and, thus, the reciprocal of the intercept is 0. In the cubic system, for example, the (100) plane intercepts the x axis at a and is parallel to the yz plane, that is, it intercepts the y and z axes at ∞.

Depending on the crystal system, two or more planes may belong to the same family of planes. In the cubic system, for example, the six sides of the unit cell has Miller indices of

These planes are equivalent except for our arbitrary choice of axis labels and directions, or, alternatively, on how we view the orientation of the unit cell in space. They have the same arrangement of atoms and, thus, the properties in these planes are identical. These planes are described collectively as {100} planes where the Miller indices are enclosed within curly brackets. In general, a family of equivalent planes with Miller indices h, k, and l is written {hkl}.