Q.Derive Bragg's equation.

What is a Unit Cell?

Imagine you're tiling a bathroom floor. You pick up one tile — a square, maybe. If you know exactly what that one tile looks like — its shape, its colour, its pattern — and you know how to slide it left, right, up, and down, you can recreate the entire floor without ever looking at the whole thing again. That single tile is the repeat unit of the floor.

A crystal is just a three-dimensional version of that tiled floor. Atoms, ions, or molecules arrange themselves in a repeating pattern that extends in all three directions. The unit cell is the smallest three-dimensional "tile" that, when stacked side by side in every direction, builds the entire crystal.

The Precise Statement

A unit cell is the smallest repeating three-dimensional portion of a crystal lattice. When translated (shifted) by integer multiples of its edge lengths along its three axes, it reproduces the entire crystal structure without gaps or overlaps.

In other words: if you know the contents of one unit cell — which atoms are inside it, and where they sit — you know the whole crystal. The entire solid is just that cell, repeated endlessly.

Key Features of a Unit Cell

A unit cell is defined by three edge lengths — usually labelled $a$, $b$, and $c$ — and three angles between them — $\alpha$, $\beta$, and $\gamma$. These six numbers are called the lattice parameters.

• $a$, $b$, $c$: the lengths of the cell edges
• $\alpha$: angle between edges $b$ and $c$
• $\beta$: angle between edges $a$ and $c$
• $\gamma$: angle between edges $a$ and $b$

Depending on these parameters, unit cells fall into seven crystal systems (cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, triclinic). The simplest and most symmetric is the cubic unit cell, where $a = b = c$ and $\alpha = \beta = \gamma = 90^\circ$.

Types of Unit Cells (for Cubic System)

Even within a cubic cell, atoms can be arranged in different ways. The three most common types are: