Question

Which of the three-dimensional primitive lattices has a unit cell where none of the internal angles is \(90^{\circ}\) ? (a) Orthorhombic, (b) hexagonal, (c) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

Short answer

The correct answer is (e) both rhombohedral and triclinic lattices, as they are the only ones among the given lattice types where none of the internal angles is equal to 90°.

Step-by-step solution

Orthorhombic lattice

In an orthorhombic lattice, all three axes are mutually perpendicular, which means that the internal angles are exactly 90°. So, an orthorhombic lattice does not match the required conditions.

Hexagonal lattice

In a hexagonal lattice, the angle between the a and b axes is 120°, while the angle between the a and c axes and between the b and c axes is 90°. By definition, the hexagonal lattice has at least one angle equal to 90°, so it doesn't satisfy the conditions.

Rhombohedral lattice

In a rhombohedral lattice, the angles between each pair of axes are equal and not 90° (they are equal to α). As this lattice has all internal angles not equal to 90°, it meets the required conditions. Therefore, it could be one of the answers.

Triclinic lattice

In a triclinic lattice, all three axes are different in length and the internal angles between them are α, β, and γ, where none of these angles is equal to 90°. Therefore, this lattice also meets the required conditions and could be another answer.

Both rhombohedral and triclinic

Since both rhombohedral and triclinic lattices meet the required conditions, the correct answer is (e) both rhombohedral and triclinic.

Key concepts

Primitive Lattices

Primitive lattices are the most basic types of crystal lattices, making up the foundation of crystal structures. They consist of points that are arranged in a periodic pattern in three dimensions. Each of these points can represent a unit cell, and in primitive lattices, there is exactly one lattice point per unit cell. This simplicity makes primitive lattices an ideal starting point for understanding the more complex structures found in crystallography.
In three-dimensional space, there are several primitive lattice types, each defined by different arrangements and angles between their axes:

• Cubic lattice: All sides are equal, and all angles are 90°.
• Tetragonal lattice: Two sides are equal, one is different, and all angles are 90°.
• Orthorhombic lattice: All sides are different, but all angles are 90°.
• Rhombohedral lattice: All sides are equal, but none of the angles is 90°.
• Triclinic lattice: All sides and angles are different, with no 90° angles.

These variations allow each lattice to serve distinct roles in forming different crystal structures. Both rhombohedral and triclinic lattices stand out among primitive lattices as their unit cell angles deviate from 90°, making them unique in occupancy and symmetry.

Crystal Structures

Crystal structures represent the orderly and repetitive arrangement of atoms or molecules in a crystal. Each structure is formed from a combination of a lattice and a basis. The lattice provides the geometric framework, while the basis consists of the specific atoms or molecules that repeat within each unit cell.
Several factors influence how atoms are arranged in a crystal structure:

• Chemical bonding: The type of chemical bonds affects the distance and arrangements of atoms within the lattice.
• Electron cloud distribution: The shape and extent of electron clouds around atoms can influence how closely the atoms pack together.
• Temperature and pressure: These conditions can alter atomic arrangements, sometimes leading to different crystal structures for the same substance (a phenomenon known as polymorphism).

The study of crystal structures is vital as they directly influence the physical properties of materials, like hardness, conductivity, and optical characteristics. Therefore, understanding primitive lattices and their associated angles is essential for predicting material behavior.

Unit Cell Angles

The angles between the edges of a unit cell are known as unit cell angles, and they play a crucial role in defining the geometry of a crystal lattice. These angles can significantly affect the symmetry and properties of the crystal.
In many common crystal systems, specific unit cell angles are employed:

• Cubic systems: All angles are fixed at 90°.
• Hexagonal systems: Two angles are 90° and one is 120°.
• Triclinic systems: Every angle is different and not equal to 90°.

Unit cell angles not being 90° adds complexity to crystal systems like the rhombohedral and triclinic:

• The rhombohedral lattice features equal axial lengths, with all angles uniformly greater than or less than 90°.
• The triclinic lattice lacks symmetry since all axes and angles are different, including angles not equal to 90°.

This variance in unit cell angles allows for diverse structural possibilities, influencing material properties and potential applications in various fields.