Integer representing the order of the reflection.
Calculated automatically.
Perpendicular distance between adjacent atomic planes.
Wavelength of the incident beam (e.g. X-rays).
22.64
°45.29
°8.051
keV0.77
Å1.54
ÅBragg's Law was formulated by William Lawrence Bragg and William Henry Bragg in 1913, a discovery for which they were awarded the Nobel Prize in Physics in 1915. It describes the fundamental conditions under which an X-ray, electron, or neutron beam interacts with the periodic structure of a crystal to produce constructive interference.
Unlike classical geometric optics, the Bragg angle () is not measured from the surface normal, but from the crystallographic plane itself (it is a grazing angle).
Constructive interference (and thus, the observation of a diffraction peak) occurs only when the optical path difference between rays scattered by adjacent atomic planes is equal to an integer multiple of the incident wavelength.
Bragg's Equation
The calculator allows you to solve Bragg's equation by isolating the variable of interest:
Solving for Spacing (d)
Solving for Angle (θ)
Analytical Consideration: When calculating , the equation reveals the non-linearity of diffraction. Small changes in at high angles produce significantly larger angular shifts in the spectrum (higher dispersion), which makes high-angle peaks crucial for accurate unit cell determination. The relative error in d is proportional to cot(θ), which tends to zero as θ → 90°, maximizing precision.
In X-ray diffraction, the wavelength (Å) is inversely related to the photon energy (keV) through the Planck-Einstein approximation:
Standard example: The characteristic emission line of Copper (used in most powder diffractometers) has Å, which corresponds to a photon energy of ~8.05 keV.
Since the mathematical sine function is bounded between -1 and 1, Bragg's Law imposes an unavoidable physical constraint:
If the wavelength used is greater than twice the interplanar spacing of the crystal (), it is mathematically and physically impossible to observe diffraction for that plane. For an order n, the condition generalizes to .
Cu Kα: 1.5406 Å
Mo Kα: 0.7107 Å
Co Kα: 1.78897 Å
Ag Kα: 0.5594 Å
1 Å = 10⁻¹⁰ m
1 nm = 10 Å
1 pm = 10⁻² Å