Bragg's Law and Diffraction Geometry

Bragg's Law is a fundamental equation that describes the conditions under which X-rays are diffracted by the periodic atomic planes within a crystalline solid. It is essential to the interpretation of X-ray diffraction (XRD) data and underpins many techniques used in the structural analysis of materials.

In conjunction with the geometric arrangement of incident and diffracted beams, Bragg’s Law enables the determination of crystallographic information such as interplanar spacing, lattice parameters, and crystal structure.

Bragg’s Law

Bragg’s Law is given by the equation: nλ=2dsin⁡θ

Where:

• n is the order of reflection (an integer),
• λ is the wavelength of the incident X-rays,
• d is the spacing between adjacent crystallographic planes,
• θ is the angle between the incident beam and the plane of reflection (the Bragg angle).

This relation is derived from the condition for constructive interference of X-rays scattered from adjacent atomic planes. When the path difference between reflected X-ray beams is an integer multiple of the wavelength, the waves interfere constructively, producing an observable diffraction peak.

Application in X-ray Diffraction (XRD)

Bragg’s Law is utilized in X-ray diffraction techniques to extract structural information from crystalline materials. When the Bragg condition is met for a specific set of lattice planes, a peak appears in the diffraction pattern. By analyzing these peaks, a wide range of material properties can be assessed.

Common Applications

• Phase Identification: Comparing measured peak positions to known reference patterns allows identification of crystalline phases.
• Lattice Parameter Determination: Precise measurements of diffraction angles enable calculation of unit cell dimensions.
• Crystallite Size and Microstrain: Broadening of diffraction peaks can be analyzed using the Scherrer equation and related methods.
• Texture and Orientation Analysis: Intensity variations reveal preferred crystallographic orientations within polycrystalline samples.
• Residual Stress Measurement: Shifts in diffraction peak positions can indicate strain in the crystal lattice.