Chapter 12, Problem 12.81P

Interpretation Introduction

Interpretation:

The specific difference in the position of spheres that gives a crystal structure based on the face-centered cubic unit cell less empty space than one based on the body-centered cubic unit cell is to be determined.

Concept introduction:

Crystal structure or lattice is the three-dimensional representation of atoms and molecules arranged in a particular manner. The unit cell is the smallest part of the lattice that is repeated in all directions to yield the crystal lattice. There are 3 types of cubic unit cells as follows:

1. The simple cubic unit cell

2. Body-centered unit cell

3. Face-centered unit cell

In the cubic unit cell, atom at the corner is shared by 8 adjacent cells so the contribution of an atom at the corner is $\frac{1}{8}$, atom at the face is shared by 2 adjacent cells so the contribution of an atom at the face is $\frac{1}{2}$ and atom at the body center is shared by 1 cell so the contribution of an atom at the body center is $1$. The number of atoms present in a simple cubic, a body-centered cubic, and a face-centered cubic unit cell are 1, 2 and 4 respectively.