Читать книгу Solid State Physics - Philip Hofmann - Страница 22

Obviously, the Bragg condition will be satisfied not only for a special kind of lattice plane in a crystal, such as the hexagonal planes in an hcp crystal, but for all possible parallel planes in a structure. Thus, we need a more precise definition of the term lattice plane. It proves useful to define a lattice plane as a plane containing at least three non‐collinear lattice points of a given Bravais lattice. If it contains three points, it will actually contain infinitely many because of the translational symmetry of the lattice. Examples for lattice planes in a simple cubic structure are shown in Figure 1.9.

Figure 1.9 Three different lattice planes in the simple cubic structure characterized by their Miller indices.

Following this definition, all lattice planes can be characterized by a set of three integers, the so‐called Miller indices. We derive them in three steps:

1 We find the intercepts of the specific plane at hand with the crystallographic axes in units of the lattice vectors, for example, for the leftmost plane in Figure 1.9.

2 We take the “reciprocal value” of these three numbers. For our example, this gives .

3 We multiply the numbers obtained in this manner with some factor so that we arrive at the smallest set of integers having the same ratio. In the example given, this is not necessary as all number are already integers.

Such a set of three integers can then be used to denote any given lattice plane. Later, we will encounter a different and more elegant definition of the Miller indices.

In practice, the X‐ray diffraction peaks are so sharp that it is difficult to align and move the sample so that the incoming and reflected X‐rays lie in a plane normal to a certain crystal plane. An elegant way to circumvent this problem is to use a powder consisting of very small crystals instead of a large single crystal. This will not only ensure that some of the many crystals are oriented correctly to get constructive interference from a certain set of crystal planes, it will also automatically yield the interference pattern for all possible crystal planes.