What is the packing efficiency of a body-centered cubic unit cell?

Verified step by step guidance

1. Understand the concept of packing efficiency, which is the fraction of volume in a crystal structure that is occupied by atoms.
2. Recognize that in a body-centered cubic (BCC) unit cell, there is one atom at each corner of the cube and one atom in the center of the cube.
3. Calculate the number of atoms per unit cell in a BCC structure: 8 corner atoms each contributing 1/8th of an atom to the unit cell, plus 1 center atom, totaling 2 atoms per unit cell.
4. Determine the relationship between the atomic radius (r) and the edge length (a) of the BCC unit cell: \( \sqrt{3}a = 4r \).
5. Calculate the packing efficiency using the formula: \( \text{Packing Efficiency} = \frac{\text{Volume of atoms in the unit cell}}{\text{Volume of the unit cell}} \times 100\% \).

Key Concepts

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that reflects the symmetry and structure of the entire crystal. In the case of a body-centered cubic (BCC) unit cell, it consists of atoms located at each of the eight corners of a cube and one atom at the center. Understanding the arrangement of atoms within the unit cell is crucial for calculating packing efficiency.

Packing Efficiency

Packing efficiency is a measure of how effectively the atoms in a crystal structure occupy space. It is defined as the ratio of the volume occupied by the atoms in the unit cell to the total volume of the unit cell. For a body-centered cubic structure, this efficiency can be calculated using the formula: (number of atoms × volume of one atom) / volume of the unit cell.

Body-Centered Cubic (BCC) Structure

The body-centered cubic (BCC) structure is a type of crystal lattice where one atom is positioned at each corner of a cube and one atom is at the center of the cube. This arrangement leads to a specific coordination number of 8 and a unique packing efficiency. Understanding the geometry and atomic arrangement in BCC is essential for determining its packing efficiency.