Bragg's Diffraction Law
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.
Electromagnetic waves, like X-rays, are an excellent tool for learning about the structure of matter, and Bragg's diffraction law shows why. Imagine 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 add up constructively, giving intense radiation, and some add up destructively.
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 one lattice length deep, and so on. The incident wave reflects from these layers and the reflected waves then interfere. The maximum intensity happens when the extra distance travelled by the wave between two consecutive layers of atoms is a whole number of wavelengths. This is Bragg's diffraction law. By observing the maxima, we can infer the distance between the atoms.
Bragg's Equation
Bragg's equation is the condition on the angle of the incident wave for which a maximum of the diffraction pattern occurs:
nλ = 2d · sin(θ)
Where:
- n — a positive integer, the diffraction order;
- λ — the wavelength of the X-ray;
- d — the interplanar distance, i.e. the distance between consecutive layers of atoms; and
- θ — the angle of the incident X-ray (measured from the lattice planes).
The order n tells how many times the wavelength λ fits between the double interplanar distance. It must be an integer, and its maximum value is (2d) / λ.
How to Use the Bragg's Law Calculator
The calculator offers three modes:
- Find diffraction angles — enter the wavelength λ and the interplanar distance d. The calculator returns the number of observable maxima and the incidence angle θ for every diffraction order n.
- Find interplanar distance — enter the wavelength λ, the order n, and the angle θ to obtain the spacing d = nλ / (2·sinθ).
- Find wavelength — enter the interplanar distance d, the order n, and the angle θ to obtain λ = 2d·sinθ / n.
Worked Example
Let us take an X-ray of wavelength λ = 120 pm. For a crystal with an interplanar distance d = 280 pm we get 4 maxima, because n must be smaller than (2d) / λ = 4.67.
sin(θ) = nλ / (2d)
| Order (n) | sin(θ) | Angle θ |
|---|---|---|
| 1 | 0.21429 | 12.374° |
| 2 | 0.42857 | 25.377° |
| 3 | 0.64286 | 40.005° |
| 4 | 0.85714 | 59.000° |
We will see the maximum diffraction pattern at these incident X-ray angles.
Metric and American Units
The calculator supports both metric units (picometers, ångströms, nanometers, micrometers) and US/imperial units (microinches, mils, inches) for the wavelength and the interplanar distance. The diffraction angle is dimensionless and is always reported in degrees. Results for d and λ are shown in several units so you can read them in whichever system you prefer (1 Å = 100 pm = 0.1 nm ≈ 3.937 µin).
Where Does Bragg's Law Find Its Application?
Bragg's law is the foundation of X-ray crystallography, the technique used to determine the atomic and molecular structure of crystals. It has been essential in chemistry, mineralogy, materials science, and molecular biology — for instance in revealing the double-helix structure of DNA. The same principle applies to neutron and electron diffraction, and to engineered structures such as X-ray mirrors and photonic crystals.
FAQs
What is the maximum diffraction order?
Because sin(θ) cannot exceed 1, the maximum order is n_max = floor((2d) / λ). If the wavelength is larger than twice the interplanar distance, no diffraction maxima exist at all.
Is the Bragg angle measured from the surface or from the normal?
In Bragg's law the angle θ is measured between the incident ray and the lattice planes (the crystal surface), not the normal. This is the opposite convention to Snell's law, where the angle is measured from the normal.
Does Bragg's law only apply to X-rays?
No. Bragg's law applies to any wave whose wavelength is comparable to the lattice spacing, including neutrons and electrons. X-rays are the most common choice because their wavelengths (around 0.1 nm) match typical interatomic distances.
If you find crystallography interesting, you can also check our Snell's law calculator and thin lens equation calculator.