X-ray diffraction is a powerful tool for uncovering the secrets of crystal structures. Bragg's law, the cornerstone of this technique, explains how X-rays scatter off crystal planes to create unique diffraction patterns.

Understanding Bragg's law and diffraction conditions is crucial for interpreting these patterns. It allows scientists to determine atomic arrangements, measure interatomic distances, and identify unknown materials across various scientific fields.

Bragg's Law and X-ray Diffraction

Fundamentals of Bragg's Law

• Bragg's law expressed as $n\lambda = 2d \sin\theta$
  • n represents an integer
  • λ denotes the wavelength of incident X-rays
  • d signifies the interplanar spacing
  • θ indicates the angle of incidence
• Describes conditions for constructive interference of X-rays scattered by crystal lattice
• Provides fundamental relationship between incident radiation wavelength, crystal plane spacing, and diffraction angle
• Crucial for determining crystal structures through X-ray diffraction techniques
  • Allows calculation of interplanar spacings from observed diffraction patterns
• Applies to other forms of radiation with wavelengths comparable to interatomic distances (neutrons, electrons)
• Forms basis for various crystallographic techniques
  • Single-crystal X-ray diffraction
  • Powder X-ray diffraction methods

Applications and Significance

• Enables structure determination of crystalline materials
  • Proteins (hemoglobin)
  • Inorganic compounds (sodium chloride)
  • Organic molecules (caffeine)
• Facilitates material characterization in diverse fields
  • Materials science (alloys, ceramics)
  • Pharmaceuticals (drug polymorphs)
  • Geology (mineral identification)
• Supports quality control in manufacturing processes
  • Semiconductor industry (silicon wafer analysis)
  • Pharmaceutical production (drug purity assessment)
• Aids in scientific research across disciplines
  • Chemistry (reaction mechanisms)
  • Physics (novel materials)
  • Biology (protein-ligand interactions)

Derivation of the Bragg Equation

Geometric Considerations

• Begins with two parallel X-ray beams reflecting off adjacent crystal planes
• Path difference between beams must equal integer multiple of wavelength for constructive interference
• Path difference geometrically expressed as $2d \sin\theta$
  • d represents interplanar spacing
  • θ denotes angle of incidence
• Equating path difference to integer multiple of wavelength yields Bragg's equation
  • $n\lambda = 2d \sin\theta$
• Demonstrates unique diffraction patterns for each crystal structure
  • Depends on specific arrangement and spacing of atoms in lattice

Relationship Analysis

• Fixed wavelength λ: decreasing interplanar spacing d requires increasing diffraction angle θ
• Fixed interplanar spacing d: longer wavelengths result in larger diffraction angles
• Enables prediction of diffraction angles for known crystal structures
• Allows determination of unknown crystal structures from measured diffraction patterns
• Illustrates inverse relationship between d-spacing and sin θ
  • Larger unit cells produce diffraction peaks at smaller angles
  • Smaller unit cells yield diffraction peaks at larger angles

Bragg's Law for Constructive Interference

Conditions for Diffraction

• Constructive interference occurs when path difference equals integer multiple of wavelength
• Diffraction condition met when incident angle θ and interplanar spacing d combine to satisfy Bragg's law
• Diffraction peaks occur at specific angles (Bragg angles) for given set of lattice planes
• Intensity of diffraction peaks influenced by structure factor
  • Depends on arrangement and types of atoms within unit cell
• Systematic absences in diffraction patterns explained by applying Bragg's law to specific crystal symmetries and lattice types
• Width of diffraction peaks related to crystal size and perfection
  • Broader peaks indicate smaller or less perfect crystals
• Bragg's law predicts positions of diffraction spots in reciprocal space

Practical Applications

• Enables determination of crystal symmetry and space group
  • Absence of certain reflections indicates specific symmetry elements
• Allows calculation of unit cell parameters
  • a, b, c (lattice constants)
  • α, β, γ (angles between axes)
• Facilitates phase identification in mixtures
  • Comparison of observed patterns with database of known structures
• Supports quantitative analysis of crystalline phases
  • Relative peak intensities correlate with phase abundance
• Enables study of crystal orientation in textured materials
  • Preferred orientation in thin films or rolled metals

Reciprocal Lattice and Diffraction Conditions

Reciprocal Lattice Concept

• Mathematical construct representing Fourier transform of real crystal lattice in reciprocal space
• Each reciprocal lattice point corresponds to set of parallel planes in real lattice
  • Distance from origin inversely proportional to interplanar spacing
• Reciprocal lattice vectors a*, b*, and c defined in terms of real lattice vectors a, b, and c
  • Form orthogonal basis in reciprocal space
• Simplifies analysis of diffraction patterns
  • Directly relates to observed diffraction spots
• Essential for advanced crystallographic techniques
  • Structure factor calculations
  • Phase determination methods

Ewald Sphere and Diffraction Geometry

• Ewald sphere construction provides geometric interpretation of Bragg's law and diffraction conditions
• Sphere radius equals 1/λ (inverse of X-ray wavelength)
• Diffraction occurs when reciprocal lattice points intersect Ewald sphere surface
• Rotation of crystal during experiment causes reciprocal lattice to rotate through Ewald sphere
  • Generates series of diffraction spots
• Enables visualization of accessible reflections for given experimental setup
• Helps in planning diffraction experiments
  • Determining required crystal rotations
  • Optimizing detector positioning