Question

What is the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice? (a) 1 , (b) 2 , (c) 3 , (d) 4 , (e) 5 .

Short answer

In a face-centered cubic (FCC) lattice, the minimum number of atoms contained in a unit cell is 4. This is calculated by considering the contributions from corner atoms (\(8\times\frac{1}{8}=1\)) and face-centered atoms (\(6\times\frac{1}{2}=3\)), resulting in a total of \(1+3=4\) atoms. Therefore, the correct answer is (d) 4.

Step-by-step solution

Understanding the face-centered cubic lattice

A face-centered cubic lattice has atoms located at the corners and the centers of the faces of the cubic unit cell. Since there are 8 corners in a cube and 6 faces, we have atoms located at all of those positions.

Counting the corner atoms

In an FCC unit cell, each of the corner atoms is shared with 8 adjacent unit cells. Given that a corner atom is shared equally among these cells, the contribution of each corner atom to a particular unit cell is 1/8. As there are total 8 corners in a unit cell, the contribution from the corner atoms can be calculated as follows: Contribution from corner atoms \(= 8 \times \frac{1}{8} = 1\)

Counting the face-centered atoms

The face-centered atoms are shared with only one other adjacent unit cell. Therefore, the contribution of each face-centered atom to a particular unit cell is 1/2. Since there are 6 faces in a cubic unit cell, the contribution from the face-centered atoms can be calculated as follows: Contribution from face-centered atoms \(= 6 \times \frac{1}{2} = 3\)

Calculating the total number of atoms

Total number of atoms in an FCC unit cell = Contribution from corner atoms + Contribution from face-centered atoms Total number of atoms \(= 1 + 3 = 4\) The minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice is 4. So the correct answer is (d) 4.

Key concepts

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that fully captures the symmetry and structure of the entire crystal. It acts like a tiny building block for the entire solid structure. Just like how bricks are used to build walls, unit cells stack in 3D patterns to form the entire crystal lattice.

• Represents the crystal structure
• Contains the minimum number of atoms to exhibit full crystal symmetry
• Repeated uniformly in all directions to create the bulk crystal

Understanding the unit cell is crucial in materials science, as it allows us to comprehend the properties of the material by just studying this elemental unit.
In a face-centered cubic (FCC) crystal lattice, the unit cell contains atoms arranged at various positions, deeply influencing the material's mechanical and thermal properties.

Corner Atoms

In a cubic unit cell, atoms are located at the corners of the cube. These corner atoms play a vital role in defining the unit cell but they are not wholly contained within it. Each corner atom is shared with several adjacent unit cells.

• Located at each of the eight corners of the cube
• Shared among eight neighboring unit cells

Due to their shared nature, the corner atoms' contribution to a single unit cell is fractional. Specifically, each corner atom contributes only \(\frac{1}{8}\)th of itself to a given unit cell.
For a cubic unit cell, with its eight corners: \[\text{Contribution from corner atoms} = 8 \times \frac{1}{8} = 1\] This total contribution is crucial for determining the actual number of atoms per unit cell.

Face-Centered Atoms

Face-centered atoms are situated at the center of each face of the cubic unit cell. These are not shared as extensively as the corner atoms; instead, they are shared between only two adjacent unit cells. They heavily influence packing efficiency and density.

• Located at the centers of each of the six cube faces
• Shared between two neighboring unit cells

This sharing results in each face-centered atom contributing \(\frac{1}{2}\) of itself to one unit cell.
For the face-centered cubic lattice:\[\text{Contribution from face-centered atoms} = 6 \times \frac{1}{2} = 3\] Thus, face-centered atoms significantly add to the unit cell's total atomic count, impacting the material properties like durability and melting point.

Cubic Unit Cell

The cubic unit cell is one of the most common types of unit cells and includes simple cubic, body-centered cubic, and face-centered cubic variations. Each type has distinct properties based on the arrangement and sharing of its atoms.
In a face-centered cubic (FCC) unit cell:

• Atoms are situated at each of the eight corners.
• Additional atoms are at the center of each face of the cube.

The unique arrangement results in a high packing efficiency of about 74%, making FCC structures denser than many other structures.
The total number of atoms contributed by both corner and face-centered atoms in an FCC unit cell: \[\text{Total atoms} = 1 \text{ (corner atoms)} + 3 \text{ (face-centered atoms)} = 4\]This detailed arrangement isn't just about counting atoms; it explains how materials with FCC lattices often possess desirable traits such as high ductility and strength.