# Crystal Planes and Miller Indices

*Crystal planes* come from the structures known as crystal lattices. These lattices are three dimensional patterns that consist of symmetrically organized atoms intersecting three sets of parallel planes. These parallel planes are "crystal planes" and are used to determine the shape and structure of the unit cell and crystal lattice. The planes intersect with each other and make 3D shapes that have six faces. These crystal planes define the crystal structure by making axes visible and are the means by which we can calculate the *Miller Indices*.

### Introduction

The orientation of a surface or a crystal plane may be defined by considering how any plane intersects the main crystallographic axes of the solid. The application of a set of rules leads to the assignment of the Miller Indices, (hkl). These are a set of numbers that may be used to identify the plane or surface.

To calculate Miller Indices, follow these steps:

- Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions, then:

- Take the reciprocals
- Clear fractions
- Reduce to lowest terms

For example, if the x-, y-, and z- intercepts are 3,1, and 6, the Miller indices are calculated as follows:

- Take reciprocals: 1/3, 1/1, 1/6
- Clear fractions {multiply by 6}: 2, 6, 1
- Reduce to lowest terms (already there)

### General principles to remember about Miller Indices

- If a Miller index is zero, the plane is parallel to that axis.
- When a Miller index is smaller, the plane is more parallel to that axis.
- When a Miller index is bigger, the plane is more perpendicular to that axis.

### References

- Petrucci, et al. General Chemistry Principles & Modern Applications. 9th ed. Upper Saddle River, NJ: Pearson Prentice Hall, 2007

### Contributors

- Timothy Ulleseit (UCD)