Free Bragg's Law Calculator

What Is Bragg’s Law?

Bragg’s law, also known as the Bragg equation, describes the condition for constructive interference of X‑rays scattered by the periodic atomic planes in a crystal. This online Bragg’s law calculator acts as a dedicated crystal lattice calculator and X‑ray diffraction calculator: you can quickly compute the diffraction angle or interplanar spacing when the other parameters are known.

How Bragg’s Diffraction Works

When an X‑ray beam hits a crystalline sample, each atom scatters the incident radiation. The scattered waves interfere; where the path difference between rays reflected from adjacent lattice planes equals an integer multiple of the wavelength, a strong diffraction peak appears. The condition for such constructive interference is given by the Bragg equation:

$$n\lambda = 2d \sin\theta$$

where:

• $n$ – a positive integer called the order of the reflection,
• $\lambda$ – the wavelength of the X‑rays (typically expressed in pm or Å),
• $d$ – the interplanar spacing (distance between consecutive atomic layers),
• $\theta$ – the angle between the incident beam and the crystal plane (the Bragg angle).

Because the path difference must be a whole number of wavelengths, the order $n$ is limited by the ratio $2d/\lambda$. The maximum possible integer order is $n_{\max} = \lfloor 2d/\lambda \rfloor$.

Using the Calculator: A Worked Example

To see how the Bragg equation is applied, consider an X‑ray source with a wavelength of 120 pm and a crystal whose interplanar distance is 280 pm. The largest integer order is $n_{\max} = \lfloor (2 \times 280) / 120 \rfloor = 4$. The Bragg law gives the scattering angle for each order:

$$\theta = \arcsin\!\left(\frac{n\lambda}{2d}\right)$$

Substituting the numbers:

$n$	$\theta$ (degrees)
1	$\arcsin(120/560) \approx 12.374^\circ$
2	$\arcsin(240/560) \approx 25.377^\circ$
3	$\arcsin(360/560) \approx 40.005^\circ$
4	$\arcsin(480/560) \approx 58.997^\circ$

These four angles correspond to the observable diffraction maxima. The calculator can compute any of these unknowns—given $n$, $d$, and $\lambda$ it returns $\theta$, or given $\theta$ and $\lambda$ it solves for $d$.

Practical Applications of Bragg’s Law

Bragg’s diffraction law is a cornerstone of X‑ray crystallography. Scientists use it to determine the atomic structure of materials ranging from simple metals (e.g., body‑centered cubic chromium) to complex biological macromolecules. The technique was instrumental in solving the double‑helix structure of DNA. Beyond X‑rays, the same principle applies to electron diffraction in transmission electron microscopy (TEM), where electrons are treated as matter waves (de Broglie relation). In the electron case, the wavelength depends on the electron momentum, but the Bragg condition $n\lambda = 2d\sin\theta$ remains valid, allowing the study of crystal lattices at the nanoscale.

Whether you are a student learning crystallography or a researcher characterizing a new material, this online Bragg’s law calculator simplifies the routine evaluation of the Bragg equation and helps you convert between diffraction angles and lattice spacings.