Bragg's Law

Bragg's law describes the condition under which waves scattered by regularly spaced planes in a crystal reinforce one another. It relates the wavelength of the incident radiation to the separation of the planes and the angle of incidence through $n\lambda=2d\sin\theta$. A fundamental relation in crystallography, it provides the geometric basis for interpreting X-ray diffraction patterns and determining distances within crystalline materials. The apparent “reflection” from crystal planes represents the combined effect of scattering and constructive interference, rather than ordinary reflection from a material surface. (iucr.org)

Equation and angle conventions

The conventional form is

where $\lambda$ is the wavelength, $d$ is the perpendicular distance between adjacent planes in the chosen family, $n$ is a positive integer specifying the diffraction order, and $\theta$ is the angle between the incident beam and those planes. It is not the angle measured from the plane normal. The angle between the forward incident-beam direction and the diffracted beam is $2\theta$, which is the scattering angle commonly reported in diffraction measurements. (iucr.org)

For a specified wavelength and diffraction order, increasing the plane spacing decreases the Bragg angle. Conversely, longer wavelengths require larger angles for the same spacing. Since $\sin\theta\leq1$, a solution requires $n\lambda\leq2d$. These relationships follow directly from the equation and explain how diffraction angles encode structural length scales. (iucr.org)

Geometric derivation

Consider two parallel incident rays scattered toward the same outgoing direction by neighboring crystal planes. The ray interacting with the deeper plane travels an additional distance on both its incoming and outgoing paths. Each contribution is $d\sin\theta$, giving a total path difference

The waves reinforce one another when this difference equals a whole number of wavelengths. Substituting $\Delta=n\lambda$ gives Bragg's law. Extending the construction to many equally spaced planes produces coherent reinforcement at the same angular condition. (iucr.org)

The planes are a geometric representation of the periodic arrangement of atoms, not separate mirrors embedded in the crystal. The mirror-like construction makes the diffraction directions easy to visualize, but the physical signal results from waves scattered throughout the material. It therefore does not imply that a crystal surface must coincide with the diffracting planes. (journals.iucr.org)

Crystal planes and reciprocal space

Families of lattice planes are labeled by Miller indices, conventionally written $(hkl)$. Their spacing, $d_{hkl}$, is related to the size and shape of the unit cell. Higher-order diffraction from a plane family of spacing $d$ can be described equivalently as first-order diffraction from planes of spacing $d/n$. Consequently, crystallographic practice commonly absorbs the order into the reflection indices and writes

This convention avoids treating the order and the indexed spacing as independent quantities. (iucr.org)

The same condition has a reciprocal-lattice description. Using wavevectors of magnitude $2\pi/\lambda$, define the scattering vector as $\mathbf q=\mathbf k_{\mathrm{out}}-\mathbf k_{\mathrm{in}}$. Diffraction occurs when $\mathbf q$ coincides with a reciprocal-lattice vector. Because $|\mathbf q|=4\pi\sin\theta/\lambda$, this condition reproduces Bragg's equation. The Ewald sphere provides a geometric construction for identifying the reciprocal-lattice points accessible at a given wavelength and crystal orientation. (dictionary.iucr.org)

Peak positions and intensities

Bragg's law determines geometrically allowed diffraction positions, but it does not determine their intensities. The arrangement and scattering strengths of the atoms within the unit cell enter through the structure factor. In the single-scattering approximation, the measured intensity is proportional to its squared magnitude, with additional experimental corrections. Contributions from different atoms can cancel, producing systematic absences even where the geometric diffraction condition is satisfied. (iucr.org)

This distinction is essential in X-ray crystallography. Peak positions constrain lattice geometry; intensities provide information about the electron density within the cell. Reconstructing that distribution uses Fourier methods and requires phase information as well as amplitudes. Bragg's law alone therefore cannot solve a crystal structure or resolve the phase problem. (iucr.org)

Experimental use and limitations

With a known wavelength, a measured peak position gives

for an indexed first-order reflection. As an illustrative calculation, radiation of wavelength $1.54$ ångströms producing a peak at $2\theta=30^\circ$ corresponds to $d\approx2.98$ ångströms. The angle substituted into the equation is $15^\circ$, not $30^\circ$.

In powder diffraction, line-profile analysis supplements peak-position measurements. Real diffraction peaks have finite widths: crystalline-domain size, distributions of lattice strain, wavelength bandwidth, and instrumental conditions can all contribute. Separating these effects requires models beyond the Bragg equation; a peak width cannot generally be interpreted as a direct measurement of one specimen property. (journals.iucr.org)

The simple relation also omits corrections arising from refraction. Such corrections can matter when precise diffraction geometry is required, particularly at shallow incidence angles. They distinguish the ideal geometric condition from the angles measured outside the crystal. (journals.iucr.org)

Historical development

William Lawrence Bragg presented the plane-reflection interpretation to the Cambridge Philosophical Society on November 11, 1912, following the discovery of X-ray diffraction by crystals earlier that year. His account explained diffraction spots through interference of radiation associated with regularly spaced crystal planes. (journals.iucr.org)

Lawrence Bragg and his father, William Henry Bragg, subsequently developed experimental methods for analyzing crystal structures with X-rays. They jointly received the 1915 Nobel Prize in Physics for that work. Their combination of diffraction geometry, measured intensities, and structural models established X-ray crystal analysis as a method for investigating atomic arrangements. (nobelprize.org)