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ssss1 Faces with different Weiss parameters on an orthorhombic crystal.

ssss1 (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.

Miller indices

(hkl)

We can use the example of the general face cited in the previous section (ssss1), 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.