Bragg’s Law is an equation that describes when X-rays bouncing off the layers of atoms inside a crystal will combine to produce a strong, detectable signal. The formula is nλ = 2d sin θ, and it connects the wavelength of the X-rays, the spacing between atomic layers, and the angle at which the beam hits the crystal. Developed in 1913 by physicist William Lawrence Bragg and his father William Henry Bragg, it became the foundation of X-ray crystallography, a technique that has since been used to map the structures of everything from table salt to DNA.
The Equation and Its Variables
The law is expressed as:
nλ = 2d sin θ
- d is the distance between parallel layers of atoms in a crystal (called the interplanar spacing), measured in nanometers or angstroms.
- θ (theta) is the angle at which the X-ray beam strikes the crystal surface, known as the Bragg angle or glancing angle.
- λ (lambda) is the wavelength of the incoming X-ray beam.
- n is a positive whole number (1, 2, 3…) representing the “order” of the reflection. First-order reflections (n = 1) are the strongest and most commonly used.
If you know any three of these values, you can solve for the fourth. In practice, researchers typically know the X-ray wavelength and measure the angle, then use the equation to calculate the spacing between atomic layers.
Why It Works: Constructive Interference
Crystals are made of atoms arranged in repeating, evenly spaced layers. When an X-ray beam enters a crystal, some of it reflects off the first layer of atoms, some off the second layer, some off the third, and so on. Each reflected wave travels a slightly different distance before exiting the crystal, because the deeper layers are farther from the surface.
That difference in travel distance is the key. Waves are periodic: they have peaks and troughs. If two reflected waves arrive back at the surface perfectly in sync, with their peaks aligned, they reinforce each other and produce a bright, intense signal. This is called constructive interference. If they arrive out of sync, peaks canceling troughs, they effectively destroy each other and you detect nothing.
Bragg’s Law defines the exact geometric condition for that reinforcement. The extra distance a wave travels to reach a deeper atomic layer and bounce back equals 2d sin θ. For constructive interference to occur, that extra path must equal a whole number of wavelengths. If the path difference is 1.5 wavelengths, for instance, the waves partially cancel. Only when it is exactly 1, 2, or 3 wavelengths (the integer n) do the reflected beams combine into a detectable peak.
A Built-In Size Requirement
There is an important constraint hidden in the math. Because sin θ can never exceed 1, the equation sets a limit: the X-ray wavelength must be smaller than twice the interplanar spacing (λ ≤ 2d). If the wavelength is too long relative to the distance between atomic layers, no angle of incidence will satisfy the equation. This is why X-rays are used for crystallography in the first place. Atomic spacings in crystals are on the order of tenths of a nanometer. Visible light, with wavelengths hundreds of times larger, simply cannot resolve structures that small. X-rays, with wavelengths in the same range as interatomic distances, fit the requirement perfectly.
For reference, the interplanar spacing in a common crystal like sodium chloride (table salt) is about 0.282 nm, and typical X-ray wavelengths used in diffraction experiments fall between 0.05 and 0.25 nm.
How Crystals Reveal Their Structure
In a real experiment, a beam of X-rays with a known wavelength is aimed at a crystal. The crystal is rotated through a range of angles, and a detector records the intensity of reflected X-rays at each angle. At most angles, little or nothing is detected because the reflected waves cancel out. But at specific angles, the Bragg condition is satisfied and a sharp intensity spike appears. These spikes are called diffraction peaks or Bragg reflections.
Each peak corresponds to a particular set of atomic planes within the crystal, each with its own spacing. By cataloging all the angles that produce peaks and plugging them into Bragg’s equation, researchers can work backward to calculate a full three-dimensional map of how atoms are arranged. Modern diffractometers automate this process, sweeping through angles and recording data that software then refines into a detailed crystal structure.
Powder diffraction, one of the most widely used material characterization methods in laboratories today, applies this same principle to ground-up crystalline samples rather than single crystals. Because a powder contains tiny crystallites oriented in every possible direction, all Bragg angles are sampled simultaneously, and the resulting pattern acts like a fingerprint for the material. This lets researchers identify chemical compositions, measure particle sizes, and detect defects. Instruments can also be paired with ovens, pressure cells, or cryostats to study how crystal structures change in real time under different conditions.
From Salt Crystals to DNA
The Braggs originally used their equation to explain why the flat faces of crystals seemed to reflect X-ray beams only at certain angles. Within two years, their work earned them the 1915 Nobel Prize in Physics “for their services in the analysis of crystal structure by means of X-rays.” William Lawrence Bragg was 25 at the time, making him the youngest Nobel laureate in Physics, a record that stood for decades.
The technique they pioneered quickly expanded beyond simple crystals. By the 1950s, X-ray crystallography was being applied to biological molecules. Rosalind Franklin’s famous X-ray diffraction images of DNA provided data on four key structural characteristics that helped James Watson and Francis Crick build their double-helix model. Earlier, the same approach had been used to verify Linus Pauling’s discovery of the alpha-helix structure in proteins. Every one of these breakthroughs relied on Bragg’s Law to translate a pattern of bright spots on film into atomic-level structural information.
Today, the law remains central to fields ranging from materials science and geology to pharmaceutical development. Whenever researchers need to know how atoms are arranged inside a solid, Bragg’s equation is the starting point.