Bragg’s Law and X-ray Diffraction
Bragg’s Law is a fundamental principle in crystallography that explains how X-rays (or other radiation) are diffracted by a crystal lattice. It was derived by William Lawrence Bragg and his father William Henry Bragg in 1913, for which they shared the Nobel Prize in Physics in 1915.
The Bragg Equation
The Bragg equation is beautifully simple yet profound in its implications:
Where:
- n is an integer called the order of diffraction (1, 2, 3, …)
- λ is the wavelength of the incident radiation (typically X-rays)
- d is the interplanar spacing between crystal lattice planes
- θ (theta) is the angle between the incident beam and the crystal planes
Physical Meaning of Bragg’s Law
When X-rays interact with a crystal, they scatter off atoms in all directions. In most directions, the scattered waves cancel each other out through destructive interference. However, in specific directions determined by Bragg’s Law, the scattered waves are in phase and combine through constructive interference to form diffraction peaks.
For constructive interference to occur, the path difference between waves reflecting from adjacent crystal planes must be equal to an integer multiple of the wavelength. This is exactly what Bragg’s Law describes.
Example Calculation:
Let’s consider a copper target X-ray source (Cu Kα radiation) with wavelength λ = 1.5406 Å.
If we observe a diffraction peak at angle θ = 15° for a first-order reflection (n = 1), what is the d-spacing?
Using Bragg’s Law: d = nλ/(2sinθ) = (1 × 1.5406 Å)/(2 × sin(15°)) = 1.5406 Å/(2 × 0.2588) = 2.98 Å
Crystal Structure and Miller Indices
In crystallography, planes in a crystal lattice are described using Miller indices (h,k,l), which are three integers that define the orientation of the plane relative to the crystal axes.
For different crystal systems, the relationship between d-spacing and lattice parameters varies:
- Cubic: d = a/√(h² + k² + l²), where a is the lattice parameter
- Tetragonal: d = 1/√((h²+k²)/a² + l²/c²), where a and c are lattice parameters
- Hexagonal: d = 1/√(4(h²+hk+k²)/(3a²) + l²/c²)
- Orthorhombic: d = 1/√(h²/a² + k²/b² + l²/c²)
Structure Factors and Reflection Intensity
Not all possible reflections are observed in X-ray diffraction patterns due to the arrangement of atoms in the unit cell. The intensity of reflections is determined by the structure factor (Fhkl), which depends on:
- The type of atoms in the unit cell
- The positions of atoms in the unit cell
- The Miller indices of the diffracting plane
Different crystal structures have different reflection conditions:
| Crystal Structure | Reflection Condition | Examples |
|---|---|---|
| Simple Cubic (SC) | All reflections allowed | NaCl |
| Body-Centered Cubic (BCC) | h + k + l = even | Fe, Cr, W |
| Face-Centered Cubic (FCC) | h, k, l all odd or all even | Cu, Al, Au |
| Diamond Cubic | h, k, l all odd or all even; h+k+l=4n gives strong reflections | Diamond, Si, Ge |
Applications of Bragg’s Law
Bragg’s Law is the foundation for numerous analytical techniques in materials science and structural biology:
- X-Ray Powder Diffraction (XRD): Used to identify and characterize crystalline materials.
- Single Crystal Diffraction: Used to determine precise atomic and molecular structures.
- Protein Crystallography: Used to determine the 3D structure of proteins and nucleic acids.
- Neutron Diffraction: Similar to XRD but uses neutrons instead of X-rays, providing different structural information.
- Electron Diffraction: Uses electrons to probe crystal structures, especially for thin films and nanomaterials.
Typical X-ray Wavelengths
| X-ray Source | Wavelength (Å) | Common Use |
|---|---|---|
| Cu Kα | 1.5406 | Standard laboratory diffraction |
| Mo Kα | 0.7107 | Single crystal diffraction |
| Co Kα | 1.7890 | Iron-containing samples |
| Cr Kα | 2.2909 | Stress analysis in steel |
| Synchrotron | Variable (0.4-2.5) | High-resolution and time-resolved studies |
Practical Considerations in X-ray Diffraction
Important Tips for XRD Analysis:
- For a given d-spacing, there’s a minimum wavelength that can produce diffraction (when sinθ ≤ 1).
- Higher order reflections (n > 1) appear at larger angles for the same set of planes.
- Peak broadening can provide information about crystallite size and strain in the material.
- Texture (preferred orientation) affects relative peak intensities in powder diffraction.
- Amorphous materials produce broad “halos” rather than sharp peaks.
Advanced Concepts: The Reciprocal Lattice
The concept of the reciprocal lattice is essential for understanding X-ray diffraction more deeply. Each point in the reciprocal lattice corresponds to a set of parallel planes in the real crystal lattice.
The magnitude of the reciprocal lattice vector is inversely proportional to the d-spacing: |G| = 2π/d
When expressed in terms of the scattering vector Q = 4πsinθ/λ, Bragg’s Law takes the elegant form:
Historical Significance
The discovery of Bragg’s Law was a pivotal moment in the history of science. It not only provided a method to determine crystal structures but also confirmed the wave nature of X-rays and the periodic arrangement of atoms in crystals. The first structures solved using X-ray diffraction—NaCl by the Braggs and diamond by W.H. Bragg—opened up a new era in our understanding of materials at the atomic level.
Today, X-ray diffraction based on Bragg’s Law continues to be an indispensable tool across scientific disciplines, from materials science and metallurgy to pharmaceuticals and structural biology.
