Question

Besides the cubic unit cell, which other unit cell(s) has edge lengths that are all equal to each other? (a) Orthorhombic, \((\mathbf{b})\) hexagonal, \((\mathbf{c})\) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

Short answer

The correct answer is (c) rhombohedral, as it has three equal edge lengths (a, a, and a) and equal angles between the edges different from \(90^{\circ}\). The orthorhombic, hexagonal, and triclinic unit cells do not have equal edge lengths.

Step-by-step solution

Analyze the orthorhombic unit cell

The orthorhombic unit cell has three unequal edge lengths (a, b, and c) that are mutually perpendicular, meaning the edge lengths are not equal to each other. This makes the answer (a) orthorhombic incorrect.

Analyze the hexagonal unit cell

The hexagonal unit cell has three equal edge lengths (a, a, and a) in the basal plane with angles \(120^{\circ}\), and the vertical axis is perpendicular to the basal plane with a different length (c). Therefore, the edge lengths are not all equal to each other, making the answer (b) hexagonal incorrect.

Analyze the rhombohedral unit cell

The rhombohedral unit cell has three equal edge lengths (a, a, and a) and angles between the edges are equal and different from \(90^{\circ}\) (given by α,β, and γ). This means that the edge lengths are equal, and the answer (c) rhombohedral is a possible correct choice.

Analyze the triclinic unit cell

The triclinic unit cell has three unequal edge lengths (a, b, and c) and none of the angles between the edges are equal to \(90^{\circ}\). This means that the edge lengths are not equal to each other, making the answer (d) triclinic incorrect.

Analyze both the rhombohedral and triclinic unit cells

Based on our previous analysis of the rhombohedral and triclinic unit cells, we know that the rhombohedral unit cell has equal edge lengths, while the triclinic unit cell does not. As a result, the answer (e) both rhombohedral and triclinic is incorrect.

Final answer

Taking all the information obtained in the previous steps, we find that the only unit cell with equal edge lengths, besides the cubic unit cell, is the rhombohedral unit cell. Therefore, the correct answer is (c) rhombohedral.

Key concepts

Crystal Structures

Crystal structures are the regular patterns that atoms, molecules, or ions arrange themselves in to form solids. These structures define many of the physical characteristics of a material, like its density, hardness, and even electrical properties.

One key feature of crystal structures is the unit cell, which is the smallest repeating pattern that expresses the full symmetry of the arrangement. Imagine a crystal as a three-dimensional wallpaper where this unit cell repeats over and over again, extending in all directions.

Different types of unit cells are possible depending on the geometry, including cubic, orthorhombic, hexagonal, rhombohedral, and triclinic. These unit cells differ by the length of their edges and the angles between them. When studying crystal structures, knowing these differences helps us understand why materials behave the way they do.

Cubic Unit Cell

The cubic unit cell is one of the simplest and most symmetric forms of unit cells in crystal structures. It features equal edge lengths, noted as 'a', 'a', and 'a'. This means that all its edges are equal, creating a regular cube.

However, its simplicity also extends to the angles between the edges, which are all \(90^{\circ}\). This regularity contributes to the unique property of cubic crystals being isotropic, meaning their properties are the same in all directions. Some well-known examples of cubic unit cells include the simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC) structures.

Each of these variations has a different arrangement of atoms within the cubic framework, leading to different material properties and applications.

Unit Cell Analysis

Unit cell analysis is a method to understand the geometry and arrangement of atoms in a crystal. By examining the unit cell, we learn how atoms are organized and can predict the material's properties.

In the analysis process, we consider the lengths of the edges of the unit cell, typically denoted by 'a', 'b', and 'c', as well as the angles between these edges. Comparison with other unit cells shows whether edges are equal or angles are what's expected for particular crystal classes.

For example, the rhombohedral unit cell is characterized by equal edge lengths with angles that are not \(90^{\circ}\). This equal edge length creates a unique symmetry compared to the triclinic cell, which has both unequal edge lengths and varied angles. Through unit cell analysis, we can determine relationships and distinctions between different crystal types, leading to explanations of why certain materials exhibit unique properties.