Understanding the packing in crystal lattices

The Bravais Lattice can be considered as the building block from which all crystals can be constructed. There exist 14 different Bravais lattices which can be grouped into 7 different crystal systems. The smallest repeating unit called the unit cell between these systems can be distinguished by a set of parameters called lattice parameters. These are a,b,c (which comprise of the lengths of edges of the unit cell) and α, β, and γ (which comprise of the angles subtended between the edges). For this experiment we will be focusing on cubic systems and hexagonal system.

Simple Cubic

The simple cubic unit cell is basically a cube with an atom at each corner. Each corner atom is shared among eight unit cells. Hence, the number of atoms per unit cell of simple cubic is 1.

• Coordination number: 6
• Packing Fraction: 0.52
• Edge Length: 2R

Body Centered Cubic

The body-centered cubic unit cell has atoms at each of the eight corners of a cube (like the cubic unit cell) plus one atom in the center of the cube (left image below). Each corner atom is shared among eight unit cells. Hence the contribution of corner atoms in BCC unit cell is (1/8)*8=1. The total no. of atoms in BCC unit cell is 2.

• Coordination number: 8
• Packing Fraction: 0.68
• Edge Length: 4R/√3

Face Centered Cubic

The face centered cubic structure has atoms located at each of the corners and the centers of all the cubic faces (left image below). Each corner atom is shared among eight unit cells. Additionally, each of its six face centered atoms is shared with an adjacent unit cell.

• Coordination number: 12
• Packing Fraction: 0.74
• Edge Length: 2R√2

Hexagonal Closed Packing

The hexagonal structure of alternating layers is shifted so its atoms are aligned to the gaps of the preceding layer. The atoms from one layer nest themselves in the empty space between the atoms of the adjacent layer just like in the fcc structure. However, instead of being a cubic structure, the pattern is hexagonal. The hcp structure has three layers of atoms. In each the top and bottom layer, there are six atoms that arrange themselves in the shape of a hexagon and a seventh atom that sits in the middle of the hexagon.

• Coordination number: 12
• Packing Fraction: 0.74
• Edge Length: 2R, 2x1.633R

Summary