Question

What specific difference in the positioning of spheres 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?

Short answer

The FCC unit cell has atoms more densely packed compared to the BCC unit cell, resulting in less empty space.

Step-by-step solution

Define Unit Cells

Understand the types of unit cells mentioned. The face-centered cubic (FCC) unit cell has atoms at each corner and at the centers of all the faces of the cube. The body-centered cubic (BCC) unit cell has atoms at each corner and a single atom at the center of the cube.

Calculate the Packing Efficiency

Calculate the packing efficiency for both unit cells. For the FCC unit cell, packing efficiency is approximately 74%. For the BCC unit cell, packing efficiency is approximately 68%. Packing efficiency is the fraction of volume in a crystal structure that is occupied by constituent particles.

Compare Atomic Arrangements

In the FCC unit cell, atoms are more closely packed together due to their positions at the cube's faces, which reduces empty space. In the BCC unit cell, the atoms are less densely packed, with more empty space in between.

Key concepts

Face-Centered Cubic Unit Cell

The face-centered cubic (FCC) unit cell is a type of crystal structure that includes atoms positioned at each corner of a cube as well as at the center of each face. This arrangement allows for a high level of organization and symmetry.
Imagine placing an atom precisely at every corner of a dice and at the center of each of its six faces.
This results in a very closely packed configuration.

• In an FCC unit cell, there are 8 corner atoms (each shared among eight unit cells).
• There are 6 face atoms (each shared between two unit cells).

Consequently, the total number of effective atoms in an FCC unit cell is 4.
This dense packing reduces empty space within the crystal structure.

Body-Centered Cubic Unit Cell

The body-centered cubic (BCC) unit cell is another type of crystal structure featuring atoms at each corner of the cube and a single atom right in the center.
Think of positioning atoms at the corners of a dice and then placing one atom in the center of the dice.
In a BCC unit cell:

• There are 8 corner atoms (each shared among eight unit cells)
• There is 1 center atom that belongs entirely to the unit cell.

This results in a total of 2 effective atoms per BCC unit cell.
The arrangement in a BCC unit cell is less densely packed compared to an FCC unit cell, creating more empty space within the structure.

Packing Efficiency

Packing efficiency measures how tightly atoms are packed in a crystal structure. It indicates the fraction of space occupied by atoms compared to the total volume of the unit cell.
In mathematical terms, the packing efficiency can be calculated as:
\[\begin{equation} \text{Packing Efficiency} = \frac{\text{Volume occupied by atoms}}{\text{Total volume of the unit cell}} \times 100\text{%} \end{equation}\]
For the FCC unit cell, the packing efficiency is approximately 74%. This high packing efficiency is due to the tightly packed atoms on the faces and the corners of the cube.
The BCC unit cell has a packing efficiency of about 68%. The central atom allows for some empty spaces, resulting in slightly less efficient packing.

To sum up:

• The FCC unit cell is more space-efficient due to its higher packing efficiency.
• The BCC unit cell has a bit more empty space within its structure.