Question

In FCC lattice A, B, C, D atoms are arranged at corners, face centres, octahedral voids and tetrahedral voids respectively, then the body diagonal contains: (1) \(2 \mathrm{~A}, 2 \mathrm{D}\) (2) \(2 \mathrm{~A}, \mathrm{C}, 2 \mathrm{D}\) (3) \(2 \mathrm{~A}, 2 \mathrm{~B}, \mathrm{D}\) (4) \(2 \mathrm{~A}, 2 \mathrm{~B}, 2 \mathrm{C}\)

Short answer

Option (2) is correct: 2A, C, 2D.

Step-by-step solution

Identify atom positions in FCC lattice

In a face-centered cubic (FCC) lattice: - 'A' atoms are at the corners, - 'B' atoms are at the face centers, - 'C' atoms are in the octahedral voids, - 'D' atoms are in the tetrahedral voids.

Understand body diagonal

A body diagonal in an FCC lattice extends from one corner of the cube to the opposite corner, passing through the interior of the cube.

Calculate positions on the body diagonal

The body diagonal of an FCC lattice intersects the following atom positions: - Two 'A' atoms at the corners.- No 'B' atoms on body diagonal (all face center).- One 'C' atom at the center of the cube (octahedral void).- Two 'D' atoms in the path since tetrahedral voids are distributed within the cube.

Verify and match the options

From the calculation, the body diagonal contains: - 2 'A' atoms- 1 'C' atom- 2 'D' atomsMatched Option: (2) 2A, C, 2D.

Key concepts

Crystal Structure

Understanding the crystal structure is essential for studying material properties. Crystals are composed of atoms arranged in a highly ordered repeating pattern. One common type of crystal structure is the face-centered cubic (FCC) lattice. Here, atoms are positioned at each of the cube's corners and the centers of all the cube faces. This specific arrangement leads to multiple types of voids and unique attributes for the geometry.
The FCC lattice has key features:

• 'A' atoms at the cube's corners.
• 'B' atoms at the face centers.
• 'C' atoms in octahedral voids.
• 'D' atoms in tetrahedral voids.

Mastering this layout allows you to predict which atoms lie along specific paths, such as the body diagonal of the cube.

Body Diagonal Calculation

The body diagonal is an imaginary line that stretches from one corner of a cube to the opposite corner, crossing through the center. To determine the atoms present along this diagonal in an FCC lattice, let's break it down.
First, consider where the body diagonal starts and ends. It begins at one corner, passes through the cube's center, and exits through the opposite corner.
In an FCC arrangement:

• The diagonal intersects two corner atoms, both 'A'.
• It crosses the center of the cube where one 'C' atom resides (octahedral void).
• The diagonal also passes through positions related to tetrahedral voids, containing two 'D' atoms distributed within the cube.

Therefore, the body diagonal comprises: 2 'A' atoms, 1 'C' atom, and 2 'D' atoms. It's crucial to visualize this path within the lattice to grasp the arrangement fully.

Octahedral and Tetrahedral Voids

Voids in crystal structures are spaces where atoms are not packed. Understanding voids helps in explaining how atoms and ions pack together within materials.
In an FCC lattice, we encounter:

• Octahedral Voids (C): These are hollow spaces forming at the body center and edge centers of the cube. Every FCC unit cell has one octahedral void located at the cube's center.
• Tetrahedral Voids (D): Smaller than octahedral voids, these occur when four atoms form a tetrahedron, creating a void at the center. Within an FCC unit cell, there are eight tetrahedral voids positioned strategically within the lattice.

Recognizing the presence and location of these voids aids in identifying how atoms like 'C' (octahedral void atoms) and 'D' (tetrahedral void atoms) align along specific lattice paths like the body diagonal.
This deeper understanding of voids and how they affect lattice structure solidifies your grasp on finding the specific atoms along unique geometric features like the body diagonal.