Skip to main content
Ch.12 - Solids and Modern Materials
Chapter 12, Problem 104

A face-centered tetragonal lattice is not one of the 14 three-dimensional lattices. Show that a face-centered tetragonal unit cell can be redefined as a body-centered tetragonal lattice with a smaller unit cell.

Verified step by step guidance
1
Step 1: Understand the structure of a face-centered tetragonal (FCT) lattice. In an FCT lattice, atoms are located at each corner of the tetragonal unit cell and at the centers of each face of the cell.
Step 2: Visualize the transformation. Consider the face-centered atoms on the top and bottom faces of the FCT lattice. These atoms can be thought of as forming a new layer that is halfway between the top and bottom layers of the original cell.
Step 3: Redefine the unit cell. By considering the face-centered atoms as part of a new layer, you can redefine the unit cell to be smaller. This new unit cell will have atoms at each corner and one atom in the center, forming a body-centered tetragonal (BCT) lattice.
Step 4: Compare the two lattices. In the BCT lattice, the body-centered atom is equivalent to the face-centered atoms of the original FCT lattice, but the unit cell is smaller, effectively reducing the volume of the unit cell.
Step 5: Conclude the transformation. By redefining the unit cell in this way, the face-centered tetragonal lattice is shown to be equivalent to a body-centered tetragonal lattice with a smaller unit cell, thus explaining why FCT is not one of the 14 Bravais lattices.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Lattice Structures

Lattice structures are the arrangements of points in space that define the periodicity of a crystal. In three-dimensional crystallography, there are 14 distinct Bravais lattices, which categorize the different ways atoms can be arranged in a crystal. Understanding these structures is essential for analyzing how different unit cells relate to one another.
Recommended video:
Guided course
00:49
Lattice Energy

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that, when repeated in three dimensions, creates the entire lattice structure. The dimensions and angles of the unit cell define the symmetry and properties of the crystal. In the context of the question, recognizing how a face-centered tetragonal unit cell can be redefined in terms of a body-centered tetragonal unit cell is crucial for understanding the relationship between these two lattice types.
Recommended video:
Guided course
01:27
Simple Cubic Unit Cell

Body-Centered vs. Face-Centered

Body-centered and face-centered lattices refer to the positions of atoms within the unit cell. In a body-centered tetragonal lattice, there is one atom at each corner and one atom at the center of the cell, while in a face-centered tetragonal lattice, there are atoms at each corner and at the center of each face. The ability to redefine a face-centered tetragonal lattice as a body-centered one with a smaller unit cell involves understanding how these atomic arrangements can be manipulated to fit different lattice classifications.
Recommended video:
Guided course
00:40
Body Centered Cubic Example