href="#ulink_67d207eb-4fb2-5663-b4b9-61ba8accbfa6">Figure 4.21 (a) The darkened front crystal face possesses the Weiss parameters: (1 : ∞ : ∞). (b) The face outlined in green possesses the Weiss parameters (1 : 1 : ∞).
Having begun to master the concepts of how Weiss parameters can be used to represent different sets of crystal planes with different sets of relationships to the crystallographic axes, students are generally thrilled to find that crystal planes are commonly referenced, not by Weiss parameters, but instead by Miller indices.
4.6.5 Miller indices
The Miller indices of any face or set of planes are the reciprocals of its Weiss parameters. They are calculated by inverting the Weiss parameters and multiplying by the lowest common denominator. Because of this reciprocal relationship, large Weiss parameters become small Miller indices. For planes parallel to a crystallographic axis, the Miller index is zero. This is because when the large Weiss parameter infinity (∞) is inverted it becomes the Miller index 1/∞ → 0.
The Miller index of any face or set of planes is, with a few rather esoteric exceptions, expressed as three integers hkl in a set of parentheses (hkl) that represent the reciprocal intercepts of the face or planes with the three crystallographic axes (a, b, and c) respectively.
We can use the example of the general face cited in the previous section (Figure 4.20), where a set of parallel planes intercepts the a‐axis at unity, the b‐axis at half unity and the c‐axis at one‐third unity. The Weiss parameters of such a set of parallel planes are 1 : 1/2 : 1/3. If we invert these parameters they become 1/1, 2/l, and 3/1. The lowest common denominator is one and multiplying by the lowest common denominator yields 1, 2, and 3. The Miller indices of such a face are (123). These reciprocal indices should be read as representing all planes that intersect the a‐axis at unity (1) and the b‐axis at one‐half unity (reciprocal is 2), and then intercept the c‐axis at one‐third unity (reciprocal is 3) relative to their respective axial ratios. Every parallel plane in this set of planes has the same Miller indices.
As is the case with Weiss parameters, the Miller indices of planes that intersect the negative ends of one or more crystallographic axes are denoted by the use of a bar placed over the indices in question. We can use the example from the previous section in which a set of planes intersect the positive end of the a‐axis at unity, the negative end of the b‐axis at twice unity and the negative end of the c‐axis at three times unity. If the Weiss parameters of each plane in the set are 1,
A simpler example is the cubic crystal shown in Figure 4.22. Each face of the cube intersects one crystallographic axis and is parallel to the other two. The axis intersected is indicated by the Miller index “1” and the axes to which it is parallel are indicated by the Miller index “0”. Therefore the six faces of the cube have the Miller indices (100), (
Figure 4.22 Miller indices of various crystal faces on a cube depend on their relationship to the crystallographic axes.
Figure 4.23 Isometric octahedron outlined in blue possesses eight faces; the form face {111} is outlined in bold blue.
Miller indices are a symbolic language that allows us to represent the spatial relationship of any crystal face, cleavage face or crystallographic plane with respect to the crystallographic axes.
4.6.6 Form indices
Every face in a form has the same general relationship to the crystallographic axes and therefore the same general Miller index, yet every face in a form has a different specific relationship to the crystallographic axes and therefore has a different Miller index. These statements can be clarified by using an example. Figure 4.23 shows the common eight‐faced isometric form called the octahedron.
Each face in the octahedron has the same general relationship to the three crystallographic axes in that each intersects the three crystallographic axes at unity. The Miller indices of each face are some form of (111). However, only the top, right front face intersects the positive ends of all three axes. The bottom, left back face intersects the negative ends of all three axes, and the other six faces intersect some combination of positive and negative ends of the three crystallographic axes. None of the faces are parallel to one another; each belongs to a different set of parallel planes within the crystal. The Miller indices of these eight faces and the set of planes to which each belongs are (111), (
