Bragg DiffractionEdit
Bragg diffraction is the phenomenon by which waves scattered from the orderly array of atoms in a crystal interfere to produce intense reflections at specific angles. When a beam with wavelength λ encounters a crystal with planes separated by a distance d, the waves reflected from successive planes can reinforce one another if the path difference is an integer multiple of λ. The condition for this constructive interference is encapsulated in Bragg's law: nλ = 2d sin θ, where θ is the angle of incidence (and reflection) and n is the order of the reflection. This simple relation underpins a powerful set of techniques for probing the arrangement of atoms in matter, most famously in X-ray crystallography and its variants.
Historically, Bragg diffraction emerged in the early 20th century from the collaboration of William Henry Bragg and his son William Lawrence Bragg. Their work demonstrated that X-rays could reveal the internal geometry of crystals, transforming chemistry, physics, and biology. The Braggs shared the Nobel Prize in Physics in 1915 for the development of X-ray crystallography as a method to determine crystal structures. From those beginnings, the theory and practice of Bragg diffraction broadened to include other radiations—neutrons and electrons—each offering different sensitivities to atomic species and magnetic structure.
History and origins
The discovery of Bragg diffraction grew out of an effort to understand why crystals produce specific, well-defined diffraction patterns when illuminated by X-rays. By measuring the angles and intensities of the diffracted beams, Bragg and Bragg showed that the pattern encodes the spacings and orientations of crystal planes. This insight provided a quantitative route to atomic-scale structure. The initial triumphs in simple salts expanded rapidly to more complex materials, and the methodology matured into a general framework for structure determination that remains foundational in modern science. See William Henry Bragg and William Lawrence Bragg for biographical context, and consider how their work relates to X-ray crystallography and Crystal structure.
Theory and principles
Bragg's law and constructive interference: The law nλ = 2d sin θ expresses the condition under which reflections from parallel lattice planes of spacing d constructively interfere. Here, θ represents the angle of incidence relative to the crystal planes, and n is a positive integer.
Planes, spacings, and Miller indices: Crystallites contain families of lattice planes characterized by Miller indices (hkl). The spacing d for a given family depends on the crystal system and lattice parameters, and it determines the angles at which Bragg reflections occur. See Miller indices and Crystal lattice for related concepts.
Reciprocal lattice and Ewald construction: The reciprocal lattice provides a convenient geometric picture of diffraction conditions. The Ewald sphere is a construction that encodes wavelength and geometry to predict where Bragg peaks arise in reciprocal space. See Reciprocal lattice and Ewald sphere.
Structure factor and intensity: The intensity of a Bragg reflection depends on the structure factor, which summarizes how the arrangement of atoms within the unit cell scatters radiation. Factors such as atomic form factors and thermal motion (the Debye-Waller factor) modulate peak heights. See Structure factor and Debye–Waller factor.
Limitations of the simple picture: Real crystals exhibit imperfections (mosaic spread, twinning, disorder) and dynamical scattering effects that can complicate interpretation. Advanced refinement methods model these features to extract accurate structural information. See Rietveld refinement for a widely used approach in powder diffraction.
Experimental methods
Bragg diffraction is observed with several complementary techniques, each optimized for different sample types and scientific questions.
X-ray diffraction: The most common modality uses X-rays to probe crystal structures. Methods include:
- Single-crystal X-ray diffraction, which yields detailed atomic coordinates for a well-ordered crystal. See Single-crystal X-ray diffraction.
- Powder (or Debye–Scherrer) diffraction, which analyzes polycrystalline samples by interpreting concentric rings or Debye–Scherrer patterns. See Powder diffraction.
- Laue method, which employs a broad spectrum of wavelengths to solve structures from informational diffraction patterns. See Laue method.
- Synchrotron sources and laboratory X-ray tubes expand capabilities in terms of flux, energy range, and time-resolved studies. See Synchrotron radiation.
Neutron diffraction: Neutrons interact with atomic nuclei and magnetic moments, offering sensitivity to light elements (like hydrogen) and to magnetic ordering. This makes neutron diffraction valuable for biological materials, minerals, and magnetic materials. See Neutron diffraction.
Electron diffraction: Electrons, with their short wavelength, are highly suited to very small crystals or thin samples. However, multiple scattering (dynamical diffraction) requires specialized interpretation. Techniques include selected-area electron diffraction (SAED) and electron backscatter diffraction (EBSD). See Electron diffraction.
Applications
X-ray crystallography in chemistry and biochemistry: Determination of molecular structures—ranging from small inorganic molecules to complex biomacromolecules such as proteins and nucleic acids—revolutionized fields from drug design to enzymology. Notable exemplars include elucidating the structures of enzymes, vitamins, and DNA. See X-ray crystallography and DNA.
Materials science and solid-state chemistry: Bragg diffraction characterizes crystal structures in metals, ceramics, polymers, and semiconductor materials, informing understanding of properties such as strength, conductivity, and phase transitions. See Crystal structure and Material science.
Geology and mineralogy: The identification of minerals and the characterization of phase compositions rely on diffraction patterns to determine lattice spacings and symmetry. See Powder diffraction and Mineralogy.
Phase and structure determination: In biology and chemistry, a combination of diffraction data with computational refinement (e.g., through methods like Rietveld refinement) yields quantitative models of atomic positions and occupancies, aiding in understanding function and reactivity. See Phase problem and Structure factor.
Limitations and debates
Bragg diffraction is a powerful, general technique, but it has limitations and areas of ongoing discussion: - Crystallinity and order: The method requires crystalline order on length scales comparable to the wavelength of the incident radiation. Amorphous materials or highly disordered regions yield broad or no Bragg peaks. - Phase problem: Diffraction experiments measure amplitudes but not phases, necessitating indirect methods (direct methods, molecular replacement, or heavy-atom derivatives) to reconstruct real-space structures. See Phase problem. - Model dependence and refinement: Structural interpretation relies on models refined against data; overfitting or bias can lead to incorrect conclusions if validation metrics are not carefully applied. See Rietveld refinement and the broader discussion of model validation in diffraction. - Radiation damage and sample limitations: X-ray and electron beams can alter sensitive samples, especially biological macromolecules and delicate materials, potentially biasing results. See Radiation damage.