Q.Derive Bragg's equation.

What is X-Ray Diffraction?

Imagine you're standing at the edge of a perfectly still pond and drop a pebble. Ripples spread outward in circles. Now drop two pebbles a short distance apart. The ripples from each source cross each other. Where a crest meets a crest, the wave gets taller; where a crest meets a trough, they cancel out. That interference pattern — the places where the water is calm and where it's rough — tells you exactly where the two pebbles fell.

X-ray diffraction is the same idea, but with atoms instead of pebbles and X-rays instead of water waves.

When a beam of X-rays hits a crystal, each atom in the crystal scatters the X-rays in all directions — like each atom becomes a tiny pebble dropped into the pond. But a crystal has billions of atoms arranged in neat, repeating planes. The scattered waves from all those atoms interfere with each other. At most angles, they cancel out completely. But at a few very specific angles, the waves from every atom in a plane add up constructively — crest on crest — and produce a strong reflected beam.

Those special angles are the ones that satisfy Bragg's law.

Here $\lambda$ is the wavelength of the X-rays, $d$ is the distance between adjacent atomic planes, $\theta$ is the angle between the incoming X-ray and the plane (not the normal), and $n$ is a positive integer (1, 2, 3, …) called the order of reflection.

The Geometry Behind the Law

Think of two parallel atomic planes separated by distance $d$. An X-ray beam hits the top plane at an angle $\theta$ and scatters off an atom there. Another X-ray, parallel to the first, travels a little further down to the next plane and scatters off an atom directly below.

The second ray has to travel an extra distance — down to the lower plane and back up. That extra path length is exactly $2d\sin\theta$.

For the two scattered rays to be in phase (crest on crest), this extra distance must equal a whole number of wavelengths: $n\lambda$. If it's anything else, the waves partially or completely cancel.

What Does This Tell Us?

If you know the X-ray wavelength (you do — it's a property of the source, like the $K_\alpha$ line from a copper target) and you measure the angle $\theta$ at which a strong reflected beam appears, you can calculate $d$:

That $d$ is the spacing between atomic planes in the crystal. Different sets of planes (different orientations) give different $d$ values. By measuring all the angles at which diffraction occurs, you can reconstruct the entire three-dimensional arrangement of atoms — the crystal structure.

• Single crystal: rotate it and record many sharp spots — each spot corresponds to a different set of planes satisfying Bragg's law.
• Powder sample: millions of tiny crystals oriented randomly. For any given $d$, some crystals will be at the right angle. You get cones of diffracted X-rays, which appear as rings on a detector. This is the Debye-Scherrer method.

Why X-Rays? …