Bragg's law

Bragg's law. It is a fundamental principle in solid-state physics that describes the condition for constructive interference when X-rays or other electromagnetic radiation strike a crystalline material. The law provides the critical relationship between the wavelength of the incident radiation, the spacing between atomic planes in the crystal lattice, and the angle at which intense reflected beams are produced. This relationship is the cornerstone of X-ray crystallography, a technique essential for determining the atomic structure of materials, from simple salts to complex biological macromolecules like DNA and proteins.

Statement of the law

Bragg's law is mathematically expressed as nλ = 2d sin θ, where n is an integer representing the order of the reflection, λ is the wavelength of the incident X-ray beam, d is the distance between successive atomic planes in the crystal lattice, and θ is the Bragg angle between the incident ray and the scattering planes. The condition arises from the path difference between waves scattered from adjacent planes; when this difference equals an integer multiple of the wavelength, the waves undergo constructive interference and produce a detectable signal. This simple equation connects the macroscopic measurement of diffraction angles to the nanoscopic atomic spacing within the crystal, enabling the determination of crystal structure.

Derivation

The derivation considers a monochromatic beam of X-rays incident at angle θ on a set of parallel lattice planes with interplanar spacing d. The beam is partially reflected by atoms in the first plane and the second plane. The extra distance traveled by the ray reflecting from the second plane is the sum of segments before and after reflection, which geometrically equals 2d sin θ. For constructive interference to occur, as dictated by the principles of wave optics, this path difference must be an integral multiple of the wavelength λ, leading directly to the law's equation. This derivation, while assuming specular reflection from the planes, successfully models the coherent scattering from a perfectly periodic Bravais lattice.

Applications

The primary application of Bragg's law is in X-ray crystallography, the principal method for determining the three-dimensional atomic structure of crystals. This technique was famously used by Rosalind Franklin to obtain the X-ray diffraction patterns of DNA that were crucial for the discovery of its double helix structure by James Watson and Francis Crick. Beyond biology, it is indispensable in materials science for characterizing metals, ceramics, semiconductors, and minerals, enabling the analysis of phases, crystal defects, and residual stress. The law also underpins the operation of X-ray spectrometers like the Bragg spectrometer and techniques such as X-ray topography and powder diffraction.

Limitations and extensions

Bragg's law is a kinematical diffraction theory that assumes a perfect, infinite crystal and neglects effects like absorption and multiple scattering. For highly perfect crystals like silicon used in semiconductor manufacturing, the dynamical theory of diffraction, developed by Paul Peter Ewald and others, is required to account for these interactions. The law also does not predict the intensity of diffracted beams, which depends on the arrangement of atoms within the unit cell, described by the structure factor. Extensions to other radiation types include neutron diffraction and electron diffraction, which probe different material properties like magnetic structure and surface morphology, respectively.

History

The law was formulated in 1912 by William Lawrence Bragg, building upon the discovery of X-ray diffraction by Max von Laue and his colleagues Walter Friedrich and Paul Knipping at the University of Munich. Lawrence Bragg, then at the University of Cambridge, realized the diffraction patterns could be interpreted as reflections from atomic planes, simplifying von Laue's more complex treatment. He and his father, William Henry Bragg, at the University of Leeds, immediately applied the law to determine the structures of sodium chloride, diamond, and other crystals, founding the field of X-ray crystallography. For this work, the Braggs were jointly awarded the Nobel Prize in Physics in 1915, making Lawrence Bragg the youngest Nobel laureate at the time. Category:Scattering Category:Crystallography Category:Scientific laws