Miller Index Visualizer

Understanding Miller Indices

Miller indices are a notation system in crystallography for planes in crystal (Bravais) lattices. In particular, a family of lattice planes is determined by three integers: h, k, and l.

They are written as (h k l). A negative integer is written with a bar over the number, but in digital formats, it's often written as -1.

How to Find Miller Indices

1. Determine Intercepts: Find the intercepts of the plane along the three crystallographic axes (x, y, z). Let's call these intercepts $a$, $b$, and $c$. If a plane is parallel to an axis, its intercept is at infinity ($\infty$).
2. Take Reciprocals: Calculate the reciprocal of each intercept. ($1/a$, $1/b$, $1/c$).
3. Clear Fractions: Multiply the reciprocals by their lowest common denominator to find the smallest set of integers (h, k, l).

Meaning of (h k l)

• The plane cuts the $x$-axis at $\frac{1}{h}$ of the lattice parameter $a$.
• The plane cuts the $y$-axis at $\frac{1}{k}$ of the lattice parameter $b$.
• The plane cuts the $z$-axis at $\frac{1}{l}$ of the lattice parameter $c$.

Families of Planes

Because of crystal symmetries, different planes might be crystallographically equivalent. For example, in a simple cubic lattice, the planes (100), (010), (001), (-100), (0-10), and (00-1) are equivalent by symmetry. This entire set is called a family of planes and is denoted by curly braces: {1 0 0}.