# Bragg's Law Calculator

Created by Miłosz Panfil, PhD
Reviewed by Dominik Czernia, PhD and Steven Wooding
Last updated: Nov 29, 2022

This Bragg's law calculator helps you explore Bragg's diffraction, a fundamental tool for investigating the structure of a crystal. Read on if you want to learn more about Bragg's diffraction law and Bragg's equation. The following text explains how to use the equation and where Bragg's law finds its application.

If you find crystallography interesting, you can also check our miller indices calculator.

## Bragg's diffraction law

Electromagnetic waves, like X-rays, are an excellent tool for learning about the structure of matter. Bragg's diffraction law shows why.

Think about an experiment in which an incident X-ray scatters from atoms in a material. The scattering changes its direction but keeps its frequency intact. The waves scattered from different atoms interfere with each other. Some waves add up constructively, giving intense radiation, and some add up destructively. Check the Brewster angle calculator and Snell's law calculator to learn more about the scattered and transmitted waves.

Atoms in crystals form regular lattices, and to X-rays, they seem like layers of mirrors. The first mirror is at the surface of the crystal; the second one is lattice length deep, and so on. The incident wave reflects from these mirrors. It is enough to understand what is going on if we consider only reflections from the first and second layers. These two reflected waves then interfere. If we vary the scattering angle, the result is the diffraction pattern.

The maximum intensity happens if the distance traveled by the wave between the first and second layers of atoms and back is proportional to the wavelength. This is the Bragg's diffraction law. Observing the maximum, we can infer the distance between the atoms.

## Bragg's equation

Bragg's equation is the condition for the angle of the incident wave for which the maximum of the diffraction pattern occurs. The formula is:

$n \lambda = 2 d \sin (\theta),$

where:

• $n$ – Positive integer, the order;
• $\lambda\ {\rm [m]}$ – Wavelength of the X-ray;
• $d\ {\rm [m]}$ – Interplanar distance, the distance between consecutive layers of atoms; and
• $\theta\ {\rm [rad]}$ is the angle of the incident X-ray.

The order $n$ tells how many times the wavelength $\lambda$ fits between the double interplanar distance. It is an integer, and its maximal value is $(2d) / \lambda$.

## How to use Bragg's law calculator

Evaluation of Bragg's equation is simple with our calculator. For example, let us take an X-ray of wavelength $\small \lambda = 120\ \rm pm$. For a crystal with the interplanar distance $\small d = 280\ \rm pm$ we get $4$ maxima because $n$ must be smaller than $\small (2 d) / \lambda = 4.67$.

The corresponding angles are $\small 12.374\degree (n = 1)$, $\small 25.377\degree (n = 2)$, $\small 40.005\degree (n = 3)$, $\small 59\degree (n = 4)$. We will see the maximum diffraction pattern at these incident X-ray angles.

## Where the Bragg's law finds its application?

Scientists use Bragg's diffraction law to understand the structure of crystals. This and similar methods are called X-ray crystallography. Materials studied with this method range from simple crystals (like the body-centered cubic cell of chromium) to large biological molecules. X-ray crystallography was an essential tool in understanding the structure of DNA.

The crystal structure of solids is also investigated using electrons instead of X-rays. Quantum mechanics tells us that electrons are also waves. Their wavelength depends on their momentum through the de Broglie equation. The diffraction of electrons is used, for example, in the Transmission Electron Microscope.

Miłosz Panfil, PhD
Wavelength (λ)
pm
Interplanar distance (d)
pm
Order (n)
Incidence angle (θ)
deg
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