Question

For structures consisting of identical atoms, how many atoms are contained in a simple cubic, a body-centered cubic, and a facecentered cubic unit cell? Explain how you obtained the values.

Short answer

SC: 1 atom, BCC: 2 atoms, FCC: 4 atoms.

Step-by-step solution

Understanding the Unit Cells

Identify the three types of cubic unit cells given: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC). Each has a different arrangement of atoms within the unit cell.

Atoms in a Simple Cubic Unit Cell

In a simple cubic unit cell, atoms are located only at the eight corners of the cube. Each corner atom is shared by 8 adjacent unit cells, so each corner atom contributes \frac{1}{8} of an atom to a single unit cell. Therefore, the total number of atoms in a simple cubic unit cell is: \[ 8 \times \frac{1}{8} = 1 \text{ atom.}\]

Atoms in a Body-Centered Cubic Unit Cell

In a body-centered cubic unit cell, in addition to the eight corner atoms (shared similarly as in the simple cubic), there is one atom located at the center of the cube that is not shared with any other unit cell. The total number of atoms in a BCC unit cell is the sum of the corner atoms and the center atom: \[ 8 \times \frac{1}{8} + 1 = 1 + 1 = 2 \text{ atoms.}\]

Atoms in a Face-Centered Cubic Unit Cell

In a face-centered cubic unit cell, there are eight corners and six face-centered atoms. Each corner atom is shared by 8 unit cells (like in SC), and each face-centered atom is shared by 2 unit cells. Thus, the contributions are as follows: \[ 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \text{ atoms.}\]

Key concepts

Simple Cubic Unit Cell

Let's start by understanding the Simple Cubic (SC) Unit Cell. In this type of unit cell, each atom is positioned at the corners of the cube. Since a cube has eight corners, there are eight atoms. However, each corner atom is shared among eight neighboring unit cells. This means each corner atom only contributes \frac{1}{8}\ of itself to the unit cell. So, the calculation for the number of atoms in a simple cubic unit cell goes like this:
\[ 8 \times \frac{1}{8} = 1 \text{ atom} \]
Therefore, a simple cubic unit cell contains exactly one atom. This arrangement leads to a low packing efficiency due to the large amount of empty space within the unit cell.

Body-Centered Cubic Unit Cell

Now, let's look at the Body-Centered Cubic (BCC) Unit Cell. In addition to the eight corner atoms, a BCC unit cell has one additional atom located right at the center of the cube. This central atom is exclusive to the unit cell and does not get shared with any other cells. Therefore, apart from the eight corner atoms, contributing \frac{1}{8}\ each, we add one whole atom from the center:
\[ 8 \times \frac{1}{8} + 1 = 1 + 1 = 2 \text{ atoms} \]
So, a body-centered cubic unit cell contains two atoms. This arrangement is more efficient than the simple cubic structure, providing a moderate packing density.

Face-Centered Cubic Unit Cell

The Face-Centered Cubic (FCC) Unit Cell is a bit different. It includes atoms at each of the eight corners and centers on each of the six faces of the cube. The corner atoms are shared among eight unit cells, while each face-centered atom is shared with just one neighboring cell, contributing \frac{1}{2}\ of itself:
\[ 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \text{ atoms} \]
Consequently, an FCC unit cell contains four atoms. This arrangement achieves the highest packing efficiency among the three, minimizing empty space and allowing for tight atomic packing.