Introduction

The unit cell is the fundamental building block of a crystal lattice. By repeating the unit cell in three dimensions, the entire crystal structure of a metal can be constructed.

Understanding unit cells is essential for calculating density, packing efficiency, coordination number, and predicting mechanical properties of metals.

Definition

Unit Cell

A unit cell is the smallest repeating structural unit of a crystal lattice that, when stacked in all directions, recreates the entire lattice.

It contains atoms at specific positions and reflects the symmetry of the crystal.

Components of a Unit Cell

1. Lattice Points: Positions in space where atoms, ions, or molecules are located.

2. Atoms: Can be fully inside the cell, at corners, or on faces/edges.

3. Edges and Angles: Define the size and geometry of the cell.

Types of Unit Cells

Unit cells can be classified based on geometry and atom arrangement:

1. Simple Cubic (SC)

   • Atoms at corners only.

   • Rare in metals.

2. Body-Centered Cubic (BCC)

   • Atoms at corners + one atom at center.

3. Face-Centered Cubic (FCC)

   • Atoms at corners + atoms at center of faces.

4. Hexagonal Close Packed (HCP)

   • Hexagonal prism arrangement, atoms in top, bottom, and middle layers.

Each unit cell type has a distinct number of atoms per cell, coordination number, and packing efficiency, which affects metal properties.

Atoms in a Unit Cell

Atoms may be shared between unit cells:

• Corner atom → shared by 8 cells → counts as 1/8 atom per cell

• Edge atom → shared by 4 cells → counts as 1/4 atom per cell

• Face atom → shared by 2 cells → counts as 1/2 atom per cell

• Center atom → belongs entirely to the cell → counts as 1 atom

Example:
Simple Cubic unit cell → 8 corner atoms × 1/8 = 1 atom per unit cell

Key Concepts

1. Lattice Parameter (a): Edge length of the unit cell.

2. Number of atoms per unit cell (n): Depends on sharing of atoms.

3. Volume of unit cell (V): For cubic cell → V = a³

4. Density of metal:

   $$\text{Density } (\rho) = \frac{n \times A}{V \times N_A}$$

   Where:
   n = number of atoms per cell
   A = atomic mass
   V = unit cell volume
   $N_A$ = Avogadro’s number

Importance in Engineering

• Determines metal density.

• Influences slip planes and ductility.

• Used in alloy design and mechanical property prediction.

• Basis for calculating atomic packing factor (APF) and coordination number.

Exam-Focused Points

• Unit cell = smallest repeating unit of a crystal lattice.

• Atoms in unit cell may be shared with adjacent cells.

• Simple cubic → 1 atom per cell, BCC → 2 atoms per cell, FCC → 4 atoms per cell, HCP → 6 atoms per cell.

• Lattice parameter (a) defines the size of the unit cell.

• Unit cell is used to calculate density, APF, and coordination number.

Common Exam Traps

• Confusing the number of atoms per cell.

• Forgetting that corner atoms are shared by 8 cells.

• Using wrong formula for density or APF.

• Mixing unit cell type with crystal system.

Example Competitive Exam Questions

What is a unit cell?
Answer — Smallest repeating structural unit of a crystal lattice.

How many atoms are there in a simple cubic unit cell?
Answer — 1 atom.

How many atoms are there in an FCC unit cell?
Answer — 4 atoms.

Define lattice parameter.
Answer — Edge length of the unit cell.

Which formula is used to calculate density of a metal using unit cell?
Answer — $\rho = \frac{n \times A}{V \times N_A}$

Quick Revision

• Unit cell = smallest repeating 3D unit of lattice.

• Atoms can be corner, face, edge, or center.

• Atoms per unit cell: SC = 1, BCC = 2, FCC = 4, HCP = 6

• Lattice parameter = edge length of unit cell

• Used to calculate density, APF, coordination number