Bragg Diffraction

Bragg Diffraction

X-ray diffraction is the primary tool for determining crystal structures. When X-rays scatter off parallel planes of atoms in a crystal, constructive interference occurs only at specific angles.

Bragg's Law:

$2d \sin(\theta) = n\lambda$

Where:

• d = interplanar spacing (distance between parallel crystal planes)
• θ = glancing angle (angle between the incident beam and the crystal plane)
• n = diffraction order (positive integer, usually 1)
• λ = X-ray wavelength

Interplanar Spacing for Cubic Crystals

For a cubic crystal with lattice constant a and Miller indices (h, k, l):

$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$

Common crystal planes:

• (100): d = a
• (110): d = a/√2
• (111): d = a/√3

Finding the Bragg Angle

Rearranging Bragg's law:

$\theta = \arcsin\left(\frac{n\lambda}{2d}\right)$

This is the glancing angle at which a diffraction peak (reflection) will be observed.

Example: NaCl with Cu Kα X-rays

Copper Kα X-rays have λ ≈ 1.54 Å. NaCl has a rock-salt structure with lattice constant a = 5.64 Å, giving d(100) = 2.82 Å. The first-order Bragg angle is about 15.8°.