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, (d) triclinic, (e) both rhombohedral and triclinic.

Short answer

The correct answer is (c) Rhombohedral, as it is the only unit cell other than the Cubic unit cell that has edge lengths equal to each other.

Step-by-step solution

(a) Orthorhombic Unit Cell

Orthorhombic unit cells have all angles equal to 90 degrees, but the edge lengths are different. Therefore, this unit cell does not meet the criterion.

(b) Hexagonal Unit Cell

Hexagonal unit cells have two edge lengths equal to each other, and the third one is different. They also have angles of 120 and 90 degrees. Therefore, this unit cell does not meet the criterion.

(c) Rhombohedral Unit Cell

Rhombohedral unit cells have all edge lengths equal to each other and all angles are equal, but less than or greater than 90 degrees. This unit cell meets the criterion.

(d) Triclinic Unit Cell

Triclinic unit cells have none of the edges equal in length and none of the angles are 90 degrees. Therefore, this unit cell does not meet the criterion. Based on the analysis above, the Rhombohedral unit cell is the only unit cell other than the Cubic unit cell that has edge lengths equal to each other. Thus, the correct answer is (c) Rhombohedral.

Key concepts

Crystal Systems

At the heart of crystallography lies the framework of crystal systems, which categorize crystals based on their symmetrical properties. There are seven primary crystal systems namely, cubic, tetragonal, orthorhombic, hexagonal, trigonal (sometimes interchanged with rhombohedral in descriptions), monoclinic, and triclinic. These systems help determine the external shapes of crystals and, importantly, how the atoms are arranged within each crystal.

Each system is defined by distinct characteristics:

• Angle measurements: The angles between edges can vary significantly from one crystal system to another.
• Edge lengths: The three edges, often referred to as lattice parameters, may be equal or different depending on the system.

Understanding these distinctions is essential for identifying the physical properties of minerals and other materials. For instance, the cubic system, noted for its equal edge lengths and 90-degree angles, provides the basic template for cubic crystals.

Rhombohedral Unit Cell

Amongst the seven crystal systems, the rhombohedral unit cell stands out due to its unique geometry. It is often interlinked with the trigonal system but is specifically characterized by:

• All three edge lengths being equal, similar to the cubic system.
• All internal angles being equal but deviating from right angles; generally less than or greater than 90 degrees.

This arrangement of equal edges and equal, non-90-degree angles creates a more skewed, diamond-like three-dimensional shape.

The rhombohedral unit cell can be found in a range of materials. For example, calcite and cinnabar are compounds that exhibit this type of crystal structure. In practical terms, the characteristics of the rhombohedral structure impart unique optical and physical properties, often utilized in various industries from optics to geology.

Edge Lengths in Crystals

Edge lengths in crystals serve as key descriptors that can dramatically influence both the symmetry and properties of a crystal. Here, lattice parameters, commonly denoted as \( a \), \( b \), and \( c \), are instrumental. They represent the repeating distances along the crystal's three primary axes.

In cubic crystals, these parameters are equal, which contributes to their highly symmetrical and balanced properties. However, in other systems such as orthorhombic or triclinic, these lengths vary, accounting for different symmetries and properties.

• Rhombohedral crystals: As previously noted, these exhibit equal edge lengths but not necessarily equal angles, unlike the cubic system.
• Relevance: These differences in edge lengths and angles help determine the mechanical, thermal, and optical properties of the material.

Such variations also aid in identifying and differentiating mineral compositions, which is crucial in both geological and industrial applications. Understanding edge lengths further enhances insight into crystallographic phenomena such as twinning, cleavage, and refractive indices.