Slicing the Crystal: Miller Indices

How do we describe a specific direction or a flat surface inside a crystal? We use Miller Indices $(hkl)$.

The Rules of the Slice

To find the indices of a plane:

1. Identify where the plane intercepts the $x, y,$ and $z$ axes.
2. Take the reciprocals of these intercepts.
3. Clear fractions to get the smallest set of integers.

Example: A plane that hits the x-axis at $1$, the y-axis at $1$, and is parallel to the z-axis has intercepts $(1, 1, \infty)$. The reciprocals are $(1, 1, 0)$. We call this the $(110)$ plane.

Interactive Plane Visualizer

Use the controls below to change the $h, k,$ and $l$ values. Watch how the plane "tilts" as you modify the integers.

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Insight: In XRD (X-ray Diffraction), these planes are what actually "reflect" the X-rays. The distance between these planes ($d_{hkl}$) determines the diffraction angle!