Problem 26 - Chemistry The Central Science

Question

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

Step-by-Step Solution

Answer

Only a rhombohedral unit cell has equal edge lengths, where a = b = c. Therefore, the correct answer is (c) Rhombohedral.

Step 1: 1. Check if Orthorhombic unit cell has equal edge lengths

In an orthorhombic unit cell, the cell edges are not necessarily equal in length. The angles between the edges are 90 degrees, but the edges can have different lengths. So, the orthorhombic unit cell does not meet the criteria.

Step 2: 2. Check if Hexagonal unit cell has equal edge lengths

A hexagonal unit cell has equal edge lengths a, b and, c, where a = b. However, the edge length c is not equal to a and b. So, the hexagonal unit cell does not meet the criteria either.

Step 3: 3. Check if Rhombohedral unit cell has equal edge lengths

A rhombohedral unit cell has equal edge lengths a, with a = b = c. Additionally, the angles α, β, and γ are equal but not equal to 90 degrees. Thus, a rhombohedral unit cell meets the criteria.

Step 4: 4. Check if Triclinic unit cell has equal edge lengths

In a triclinic unit cell, the edge lengths and angles are all different. The edges a, b, and c are different in length, and the angles α, β, and γ are also different from each other. So, the triclinic unit cell does not meet the criteria.

Step 5: 5. Conclusion

Based on our analysis of the different types of unit cells, only a rhombohedral unit cell has equal edge lengths. Therefore, the correct answer is (c) Rhombohedral.

Key Concepts

Rhombohedral Unit Cell

Rhombohedral unit cells are a fascinating and unique system in crystallography. Unlike many other unit cells, they have the defining feature of symmetry that is uniform across all three dimensions. Specifically, a rhombohedral unit cell is characterized by three equal edge lengths, denoted as \( a = b = c \). This means that each of the sides of the rhombohedral cell is of the same length, forming a sort of distorted cube.
Additionally, the angles between the edges of a rhombohedral unit cell, denoted as \( \alpha, \beta, \gamma \), are equal. However, these angles do not equal 90 degrees; instead, they are all the same, often less than 120 degrees. This differs significantly from, say, a cubic unit cell where all angles are right angles. The uniformity and equal angles give the rhombohedral system its distinctive shape, which can be visualized as a cube "squished" along one diagonal to make the angles acute or obtuse while keeping the edge lengths the same.
This unique symmetry makes rhombohedral unit cells an important study in mineralogy and materials science, often seen in crystals such as calcite. Understanding the properties of rhombohedral cells helps in recognizing and predicting the structure and behavior of materials formed through this crystal system.

Unit Cell Edge Lengths

In crystallography, understanding unit cell edge lengths is fundamental to characterizing the overall shape and symmetry of a crystal. The edge lengths are represented by \( a, b, \) and \( c \), corresponding to the lengths of the sides of the cell. Depending on the crystal system, these lengths can either be equal, unequal, or some combination thereof.
For a rhombohedral unit cell specifically, as covered, the edge lengths are all equal. This sets it apart from other unit cell types like the triclinic and orthorhombic systems, where the edge lengths vary. In the orthorhombic system, while angles between the edges are 90 degrees, the edge lengths \( a \), \( b \), and \( c \) are typically different.
The hexagonal unit cell provides an interesting comparison, where two of the edge lengths are equal (\( a = b \)), but the third \( c \) is different. This slight change creates an entirely new symmetry class distinct from that of the rhombohedral unit cell. Recognizing these nuances in edge lengths is integral for determining how different crystal structures form and behave.

Geometric Criteria in Crystallography

Geometric criteria in crystallography are crucial for categorizing and understanding the diverse crystal systems that exist. These criteria primarily revolve around the relationships between the edge lengths and the angles of unit cells.
Each of the seven crystal systems is defined by its unique geometric properties:

• In cubic systems, all edge lengths and angles are equal.
• In rhombohedral systems, edge lengths are equal, but the angles, though the same, are not 90 degrees.
• Other systems, like monoclinic or triclinic, do not have such symmetry, making their edge lengths and angles all different.

The geometric criteria help scientists predict physical properties of materials, such as symmetry, conductivity, and optical properties. Specific examples include understanding why diamond has such remarkable hardness due to its cubic structure, or why calcite may exhibit birefringence due to its rhombohedral structure.
By exploring the geometric criteria in crystallography, students and researchers can better grasp how microscopic arrangements of atoms translate into the macroscopic phenomena we observe, leading to advancements in material design and application.