Question

In what array (simple cubic, body-centered cubic, and face-centered cubic) do atoms pack most efficiently (greatest percent of space occupied by spheres)? Support your answer mathematically..

Short answer

The face-centered cubic array of atoms is packed most efficiently.

Step-by-step solution

Given Information

Different types of cubic symmetry are simple cubic, body-centered cubic and face-centered cubic.

Concept Introduction

To determine how close the spheres are packed in a unit cell, we will use the packing efficiency formula that can be written as:

$\mathbf{\%}\mathbf{Packing}\mathbf{}\mathbf{Efficiency}\mathbf{=}\frac{\mathbf{Volume}\mathbf{}\mathbf{of}\mathbf{}\mathbf{the}\mathbf{}\mathbf{atoms}\mathbf{}\mathbf{in}\mathbf{}\mathbf{the}\mathbf{}\mathbf{unit}\mathbf{}\mathbf{cell}}{\mathbf{Volume}\mathbf{}\mathbf{of}\mathbf{}\mathbf{the}\mathbf{}\mathbf{unit}\mathbf{}\mathbf{cell}}$

Step 3: Explanation

The difference between simple cubic, body-centered, and face-centered packing efficiency is determined by finding the number of spheres that fit in a unit cell.

So, a simple cubic has one atom per unit cell $\left(\frac{1}{8}\times 8=1\right)$because the cube has eight corners.

Body-centered has two- atoms per unit cell $\left(\frac{1}{8}\times 8+1=2\right)$because the cube has eight corners and one atom present in the middle.

Face-centered has four atoms per unit cell $\left(\frac{1}{8}\times 8+\frac{1}{2}\times 6=4\right)$because the cube has eight corners and six halves’ atoms present on the sides of the cube.

Hence, the most efficient cell is the face-centered cubic cell.