Question

Imagine the primitive cubic lattice. Now imagine grabbing opposite corners and stretching it along the body diagonal while keeping the edge lengths equal. The three angles between the lattice vectors remain equal but are no longer \(90^{\circ}\). What kind of primitive lattice have you made?

Short answer

After stretching the primitive cubic lattice along the body diagonal while keeping the edge lengths equal, the new primitive lattice formed is a rhombohedral (trigonal) lattice. This lattice type is characterized by having equal edge lengths and equal angles between the lattice vectors, which are no longer \(90^{\circ}\).

Step-by-step solution

Understanding the Body Diagonal Stretch

In a primitive cubic lattice, all angles between the lattice vectors are equal to \(90^{\circ}\) and all edge lengths are equal. When grabbing opposite corners and stretching along the body diagonal, the edge lengths remain equal but the angles between the lattice vectors change. Let's denote the edges of the lattice by \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

Determine the New Angles Between Lattice Vectors

To find out the new angles between the lattice vectors after stretching the cubic lattice, we focus on one corner where the stretching occurs. Since the stretching will cause all edge lengths to remain equal, this means that the new lattice vectors will be equal to the original ones: \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). However, the angles between these vectors (\(\vec{a}\) and \(\vec{b}\), \(\vec{b}\) and \(\vec{c}\) and \(\vec{c}\) and \(\vec{a}\)) will change and remain equal but no longer \(90^{\circ}\). From this information, we can determine the type of primitive lattice formed after stretching.

Identifying the New Primitive Lattice Type

Based on the changes in the stretching process, we find that: 1. The edge lengths remain equal. 2. The angles between the lattice vectors are equal but are no longer \(90^{\circ}\). These characteristics match the description of a rhombohedral lattice (also known as a trigonal lattice). In a rhombohedral lattice, all edge lengths are equal and all angles are equal. Therefore, after stretching the primitive cubic lattice along the body diagonal while keeping the edge lengths equal, we have made a rhombohedral (trigonal) lattice.

Key concepts

Primitive Cubic Lattice

The primitive cubic lattice is the simplest form of a crystal lattice. Imagining a 3D grid, this structure is like a cube where each lattice point lies at the corners of each cube. Each of the corners is shared by eight identical cubes, contributing equally to their structure.
Primitive cubic lattices have some distinctive properties:

• All sides, or vectors, are equal in length.
• The angles between the connecting lines or vectors are precisely 90 degrees.

The points at the corners represent identical repeating units or atoms. The defining feature is its symmetry, creating a regular and repetitive pattern throughout the structure. This helps form the most straightforward building block for more complex structures in crystallography.

Rhombohedral Lattice

A rhombohedral lattice, also known as a trigonal lattice, is a fascinating variation of the primitive cubic lattice that emerges when we stretch it diagonally by its body diagonal. As we manipulate the primitive cubic lattice by pulling the opposite corners, all edge lengths remain the same. However, the angles between the lattice vectors are no longer at right angles.
Key characteristics of a rhombohedral lattice include:

• All edge lengths are equal, maintaining the stretched symmetry.
• The angles between lattice vectors become equal but differ from 90 degrees.

The result is a lattice where the symmetry is transformed, yet still remains consistent in its angular equality, forming a rhomboidal shape. This structure allows for more versatile applications in materials science, where different physical properties can emerge from its altered lattice configuration.

Lattice Vectors

Lattice vectors are essential elements defining the framework of a crystal lattice. Each lattice point is connected to another by these vectors, defining the dimensional structure of the lattice. In a primitive cubic lattice, these vectors are represented as \( \vec{a}, \vec{b}, \) and \( \vec{c} \).
The length and orientation of these vectors determine the type and properties of the lattice:

• The magnitude of the vectors corresponds to the edge lengths of the unit cell.
• The angles between vectors describe the shape and dimensional integrity of the cell.

When the structure changes due to modifications, such as stretching, the relationships between vectors adjust accordingly to express new lattice forms like a rhombohedral lattice. In essence, lattice vectors act as the fundamental guides that manage the ongoing repetition and structure within a crystal lattice, providing insights into both its geometric and physical properties.