Crystallography & Miller Indices Reference Cheat Sheet

Crystallography connects the geometric structure of crystals to the chemical behavior of solids. This cheat sheet covers unit cells, lattice planes, Miller indices, crystal directions, and diffraction relationships used in solid-state chemistry. Students need these tools to describe atomic arrangements, identify planes in crystals, and interpret X-ray diffraction data.

It is designed as a quick reference for problem solving and lab analysis.

The core ideas are that a crystal can be represented by a repeating unit cell with lattice parameters $a$ , $b$ , $c$ , $\alpha$ , $\beta$ , and $\gamma$ . Miller indices $(hkl)$ describe crystal planes by using the reciprocals of their intercepts with the crystallographic axes. Important formulas include the cubic spacing equation $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$ and Bragg’s law $n\lambda = 2d\sin\theta$ .

Together, these concepts allow chemists to relate crystal geometry to diffraction peaks and material structure.

Key Facts

A unit cell is the smallest repeating volume that preserves the symmetry and structure of a crystal lattice.
Miller indices $(hkl)$ are found by taking the intercepts of a plane in units of $a$ , $b$ , and $c$ , taking reciprocals, and clearing fractions to the smallest integers.
A plane parallel to an axis has an infinite intercept on that axis, so its Miller index for that axis is $0$ because $\frac{1}{\infty} = 0$ .
For cubic crystals, the interplanar spacing is $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$ .
Bragg’s law for constructive X-ray diffraction is $n\lambda = 2d\sin\theta$ , where $n$ is diffraction order, $\lambda$ is wavelength, $d$ is plane spacing, and $\theta$ is the Bragg angle.
In cubic crystals, the direction $[hkl]$ is perpendicular to the plane $(hkl)$ , but this is not generally true for noncubic systems.
Negative Miller indices are written with a bar, such as $(\bar{1}01)$ , and represent intercepts on the negative side of an axis.
For cubic lattices, planes in the same family are written with braces as $\{hkl\}$ , and directions in the same family are written with angle brackets as $\langle hkl \rangle$ .

Vocabulary

Unit cell: The smallest repeating three-dimensional block that can generate the entire crystal lattice by translation.
Lattice parameters: The cell edge lengths $a$ , $b$ , and $c$ and the interaxial angles $\alpha$ , $\beta$ , and $\gamma$ that define a unit cell.
Miller indices: A set of integers $(hkl)$ used to label the orientation of a crystal plane.
Interplanar spacing: The perpendicular distance $d_{hkl}$ between adjacent parallel planes with Miller indices $(hkl)$ .
Bragg angle: The angle $\theta$ at which X-rays constructively interfere from parallel crystal planes according to $n\lambda = 2d\sin\theta$ .
Plane family: A symmetry-related set of equivalent planes written as $\{hkl\}$ in crystallographic notation.

Common Mistakes to Avoid

Using intercepts directly as Miller indices is wrong because Miller indices use reciprocals of intercepts, not the intercept values themselves.
Forgetting that a plane parallel to an axis has index $0$ is wrong because a parallel plane has intercept $\infty$ , and $\frac{1}{\infty} = 0$ .
Confusing $(hkl)$ with $[hkl]$ is wrong because parentheses label planes, while square brackets label directions.
Using $2\theta$ in Bragg’s law as if it were $\theta$ is wrong because diffraction instruments often report $2\theta$ , but $n\lambda = 2d\sin\theta$ uses the Bragg angle $\theta$ .
Assuming $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$ works for every crystal system is wrong because that simple form applies only to cubic crystals.

Practice Questions

1 Find the Miller indices for a plane with intercepts $2a$ , $3b$ , and $1c$ .
2 For a cubic crystal with $a = 4.00\ \text{Å}$ , calculate $d_{111}$ using $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$ .
3 An X-ray diffraction peak occurs at $2\theta = 40.0^\circ$ using radiation with $\lambda = 1.540\ \text{Å}$ and $n = 1$ . Find $d$ using $n\lambda = 2d\sin\theta$ .
4 Explain why the plane $(100)$ and the direction $[100]$ are related in a cubic crystal, and why that relationship may fail in a noncubic crystal.

Sign in to save

Sign in to save

Crystallography & Miller Indices Reference Cheat Sheet

Related Tools

Related Worksheets

Related Infographics

Study as Flashcards

Key Facts

Vocabulary

Common Mistakes to Avoid

Practice Questions