Читать книгу Crystallography and Crystal Defects - Anthony Kelly - Страница 14

A rectangular mesh of a hypothetical two‐dimensional crystal with mesh parameters a and b of very different magnitude is shown in Figure 1.4. Note that the parallel mesh lines OB, O′B′, O″B″, and so on all form part of a set and that the spacing of all lines in the set is quite regular; this is similar for the set of lines parallel to AB: A′B′, A″B″ and so on. The spacing of each of these sets is determined only by a and b (and the angle between a and b, which in this example is 90°). Also, the angle between these two sets depends only on the ratio of a to b, where a and b are the magnitudes of a and b, respectively. If external faces of the crystal formed parallel to O″B and to AB, the angle between these faces would be uniquely related to the ratio a : b. Furthermore, this angle would be independent of how large these faces were (see Figure 1.5). This was recognized by early crystallographers, who deduced the existence of the lattice structure of crystals from the observation of the constancy of angles between corresponding faces. This law of constancy of angle states: In all crystals of the same substance the angle between corresponding faces has a constant value.

Figure 1.4 A rectangular mesh of a hypothetical two‐dimensional crystal with mesh parameters a and b of very different magnitude

Figure 1.5 Demonstration of the law of constancy of angles between faces of crystals: the angle φ between the faces is independent of the size of the faces

The analogy between lines in a mesh and planes in a crystal lattice is very close. Crystal faces form parallel to lattice planes and important lattice planes contain a high density of lattice points. Lattice planes form an infinite regularly spaced set which collectively passes through all points of the lattice. The spacing of the members of the set is determined only by the lattice parameters and axial angles, and the angles between various lattice planes are determined only by the axial angles and the ratios of the lattice parameters to one another.

Prior to establishing a methodology for designating a set of lattice planes, it is expedient to consider how directions in a crystal are specified. A direction is simply a line in the crystal. Select any two points on the line, say P and P′. Choose one as the origin, say P (Figure 1.6). Write the vector r between the two points in terms of translations along the x‐, y‐ and z‐axes so that:

(1.2)

where a, b and c are vectors along the x‐, y‐ and z‐axes, respectively, and have magnitudes equal to the lattice parameters (Figure 1.6). The direction is then denoted as [uvw] – always cleared of fractions and reduced to its lowest terms. The triplet of numbers indicating a direction is always enclosed in square brackets. Some examples are given in Figure 1.7. Negative values of u, v and w are indicated in this figure by a bar over the appropriate index.

Figure 1.6 A vector r written as the sum of translations along the x‐, y‐ and z‐axes

Figure 1.7 Examples of various lattice vectors in a crystal

If u, v and w are integers and the origin P is chosen at a lattice point, then P′ is also a lattice point and the line PP′ produced is a row of lattice points. Such a line is called a rational line, just as a plane of lattice points is called a rational plane.

We designate a set of lattice planes as follows (see Figure 1.8). Let one member of the set meet the chosen axes, x, y, z at distances from the origin of A, B and C, respectively. We can choose the origin to be a lattice point. Vectors a, b and c then define the distances between adjacent lattice points along the x‐, y‐ and z‐axes, respectively. The Miller³ indices (hkl) of the set of lattice planes are then defined in terms of the intercepts A, B and C so that the length OA = a/h, where a is the magnitude of a; likewise, OB = b/k and OC = c/l.

Figure 1.8 The plane (hkl) in a crystal making intercepts of a/h, b/k and c/l along the x‐, y‐ and z‐axes, respectively

Thus, for example, the plane marked Y in Figure 1.9 makes intercepts on the axes of infinity, 2b and infinity, respectively. Taking the reciprocals of these intercepts gives h = 0, k = and l = 0. Clearing the fractions gives h = 0, k = 1 and l = 0. Hence, the set of lattice planes parallel to Y is designated (010). The triplet of numbers describing the Miller index is always enclosed in round brackets. Similarly, the plane marked P in Figure 1.9 has intercepts 1a, 2b and c. Therefore, taking the reciprocals of these intercepts, h = 1, k = and l = 3. Clearing the fractions, we have (216) as the Miller indices. The indices of a number of other planes are shown in Figure 1.9. Negative values of the intercepts are indicated in the Miller index notation by a bar over the appropriate index (see the examples in Figure 1.9).

Figure 1.9 Examples of various lattice planes in a crystal. The indices of the planes P and Y are discussed in the accompanying part of the text in Section 1.2

The reason for using Miller indices to index crystal planes is that they greatly simplify certain crystal calculations. Furthermore, with a reasonable choice of unit cell, small values of the indices (hkl) belong to widely spaced planes containing a large areal density of lattice points. Well‐developed crystals are usually bounded by such planes, so that it is found experimentally that prominent crystal faces have intercepts on the axes which when expressed as multiples of a, b and c have ratios to one another that are small rational numbers.⁴