What this calculator does
Bragg's law explains why a crystal scatters X-rays strongly at some angles and not at others. Waves reflecting off successive planes of atoms reinforce each other only when the extra path travelled is a whole number of wavelengths.
That condition turns a diffraction pattern into a measurement of atomic spacing, which is the foundation of X-ray crystallography. It is how the structure of DNA, proteins and countless materials has been determined.
The formula
The path difference between reflections off successive planes is twice the spacing times the sine of the angle. Setting that equal to a whole number of wavelengths and solving for the angle gives the Bragg condition.
| Term | Meaning |
|---|---|
| Interplanar spacing (d) | The distance between successive parallel planes of atoms in the crystal. |
| Diffraction order (n) | How many whole wavelengths of path difference the reflection corresponds to. |
| Bragg angle (θ) | The angle between the incoming beam and the crystal planes, not the plane normal. |
The inputs explained
| Field | What to enter |
|---|---|
| Diffraction order (n) | The diffraction order, a whole number. First order is the most commonly observed. |
| X-ray wavelength (pm) | The X-ray wavelength in picometres. Copper K-alpha radiation, widely used, is about 154 pm. |
| Interplanar spacing (pm) | The interplanar spacing in picometres. |
When to use it
Determining a crystal structure
Measuring the angles at which diffraction peaks appear gives the interplanar spacings and from those the structure.
Identifying a material
Each crystalline substance produces a characteristic set of diffraction angles, which works as a fingerprint.
Selecting a wavelength
The geometry has to be satisfiable, so the wavelength must be comparable to the spacing being probed.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
At what angles does diffraction occur for each order?
The same crystal and wavelength across successive diffraction orders.
| Diffraction order | Bragg angle (θ) | Full scattering angle (2θ) |
|---|---|---|
| n = 1 | 14.195° | 28.390° |
| n = 2 | 29.370° | 58.740° |
| n = 3 | 47.364° | 94.728° |
Questions
Why does diffraction only happen at specific angles?
Because the reflections from successive atomic planes must arrive in phase to reinforce. At any other angle the path difference is not a whole number of wavelengths and the reflections cancel out.
Why use X-rays rather than visible light?
Because the wavelength has to be comparable to the atomic spacing, which is a few hundred picometres. Visible light is thousands of times too long to resolve that structure.
Is there a limit on the order?
Yes. The sine of the angle cannot exceed one, so n times the wavelength cannot exceed twice the spacing. Beyond that no solution exists and those orders simply do not appear.
Is the Bragg angle measured from the surface or the normal?
From the planes themselves, which differs from the usual optical convention of measuring from the normal. That is why diffraction patterns are usually reported as 2θ, the total deviation of the beam.
For the photons doing the scattering, see the photon energy calculator. For scattering off electrons rather than planes, see the Compton scattering calculator.