1. Intercepts  Intercepts measure where a crystal face hits a crystal axis. The location on the axes corresponding to unit lengths is arbitrary and chosen for simplicity and convenience Axes usually radiate from the center in a right hand rule arrangement Axes pass through centers or edges

2. Intercepts are relative sizes  “The intercepts of a face have no relation to its size, for a face may be moved parallel to itself for any distance without changing the relative values of its intersections with the crystallographic axes.”  K&D p. 133

3. Miller Indices from Intercepts  “The Miller Indices of a face consist of a series of whole numbers that have been derived from the intercepts by  inverting, and if necessary  by the clearing of fractions.”  “The Miller Indices [also] express a ratio ….” K&D p133

4. Problem: find the Miller Index for a face with intercepts 2a, 2b, 2/3c 1.Invert the indices: ½ ½ 3/2 2. This is a ratio. If we multiply all terms by a constant, the ratio remains the same. Let’s multiply by 2 to clear the fractions: (1 1 3)

5. Miller Indices for horizontal and vertical faces  A face perpendicular to one axis may be considered to intersect the others at infinity.  For example, for a face perpendicular to the c-axis (aka a 3 -axis) the [positive side] index would be (001).

6. Problem: find the Miller Index for this face with intercepts ∞ a 1, ∞a 2, 1a 3 1.Invert the indices: Since 1/∞ = 0 and 1/1 = 1 we have 0/1 0/1 1/1 2. Clear fractions by multiplying through by 1 (0 0 1) The colored face is parallel to a1 and a2, meeting them only at ∞

7. Miller Indices for faces parallel to two axes  A face parallel to two axes may be considered to intersect the other at unity.  For example, for a face parallel to the a-axis and c-axis (or a 1 and a 3 ) the [positive side] index would be (010).

8. Miller Indices for faces parallel to one axis  If a face is parallel to one of the crystallographic axes, a zero “0” is used (because 1/infinity = 0)  For example, for a face parallel to the a-axis, the [positive side] Miller Index could be (011)

9. Faces that intersect axes on their negative side.  “For faces that intersect negative ends of crystallographic axes, a bar is placed over the appropriate number…. The bar represents the minus sign in a negative number