Body Centered Cubic Unit Cell: Videos & Practice Problems

The body-centered cubic (BCC) unit cell is a fundamental structure in crystallography, featuring a cube with one atom at each of its eight corners and a single atom at the center. This arrangement results in a total of two atoms per unit cell. The corner atoms are shared with adjacent cells, each contributing 1/8 to the unit cell, while the central atom is not shared. The edge length (a) of a BCC unit cell is calculated using the formula:

$a=\frac{4r}{\sqrt{3}}$

where r is the atomic radius. The BCC structure has a packing efficiency of 68% and a coordination number of 8, indicating each atom is surrounded by eight nearest neighbors. These characteristics are crucial for understanding the properties of materials with a BCC lattice structure.

Body Centered Cubic Unit Cell

The body-centered cubic (BCC) unit cell is a specific arrangement of atoms within a cubic structure, characterized by having an atom at each of the eight corners of the cube and one atom located at the center. This configuration results in a total of two atoms per unit cell. Each corner atom is shared among adjacent unit cells, contributing only \(\frac{1}{8}\) of an atom to the unit cell. Therefore, the total contribution from the corner atoms is \(8 \times \frac{1}{8} = 1\) atom, combined with the one atom in the center, yielding a total of 2 atoms in a BCC unit cell.

In terms of geometric relationships, the edge length (denoted as \(a\)) of a BCC unit cell is derived from the atomic radius (\(r\)). Unlike the simple cubic unit cell, where the edge length equals \(2r\) due to direct contact between atoms, the BCC structure introduces a gap. Consequently, the edge length is expressed as:

\[ a = \frac{4r}{\sqrt{3}} \]

This formula accounts for the arrangement of atoms within the unit cell. As the complexity of the structure increases, so does the packing efficiency, which for the BCC unit cell is approximately 68%. Additionally, the coordination number, which indicates the number of nearest neighbor atoms surrounding a given atom, is 8 in the case of the BCC unit cell. Understanding these key concepts and values is essential for studying the properties and behaviors of materials with a body-centered cubic structure.

Body Centered Cubic Example

Iron crystallizes in a body-centered cubic (BCC) unit cell, which is essential for understanding its atomic structure. In a cubic unit cell, each corner atom contributes 1/8 of an atom to the unit cell because it is shared among eight adjacent unit cells. Since there are eight corners in a cube, this results in a total contribution of:

8 corners × 1/8 atom/corner = 1 atom.

In addition to the corner atoms, a BCC unit cell contains one whole atom located at the center of the cube. Therefore, the total number of atoms per unit cell in a body-centered cubic structure is:

1 atom (from corners) + 1 atom (from the center) = 2 atoms per unit cell.

This understanding of atomic arrangement is crucial for studying the properties of iron, including its density and how it behaves under various conditions.

Tungsten possesses a body-centered cubic structure. If its density is 19.28 g/cm3, what is its radius in pm?

• A. 910 pm
• B. 316 pm
• C. 137 pm
• D. 3.18 pm

Answer: (C)

Vanadium has a body-centered cubic structure. If the atomic radius of vanadium is 134 pm, calculate the density of solid vanadium.

• A. 5.33 g/cm3
• B. 4.29 g/cm3
• C. 5.71 g/cm3
• D. 0.52 g/cm3

Answer: (C)

The edge of a body-centered cubic unit cell of an element Z was found to be 2.88 x 10-8 cm. The density of the element is 7.2 g/cm3. What is the approximate molar mass of Z?

• A. 101.7 g•mol-1
• B. 39.107 g•mol-1
• C. 10.291 g•mol-1
• D. 51.996 g•mol-1
  

Answer: (D)

Here’s what students ask on this topic:

How many identical atoms are in a body-centered (bcc) cubic unit cell?

In a body-centered cubic (bcc) unit cell, there are two atoms in total. Here's how it breaks down: At each corner of the cube, there's a fraction of an atom that belongs to the unit cell since each corner atom is shared with eight other unit cells. Since there are eight corners, you have:

8 × ¹⁄₈ = 1 atom from the corners.

Additionally, there is one full atom at the very center of the cube, which is not shared with any other unit cell. When you add the one atom from the corners to the one atom in the center, you get a total of 2 atoms in a body-centered cubic unit cell.