Bravais Lattice

The lattice constants are characteristic length scales associated with the lattice. It represents the distance between adjacent lattice points. Within each Bravais lattice, a repeating unit called the unit cell exists. The unit cell is the smallest repeating structure representing the entire crystal lattice.

Lattice constants define the size and shape of a unit cell. They refer to the lengths of the edges of a unit cell. They are typically denoted by letters “a”, “b”, and “c” for the three dimensions in a crystal lattice. The significance of lattice constants lies in their relationship, which is unique for each Bravais lattice.

Lattice constants also provide information about the angles between different edges or axes within a crystal structure. These angles are known as interaxial angles and are denoted by “α”, “β”, and “γ”. The values of the lengths and angles determine the overall dimensions and symmetry of the crystal structure.

There are 14 types of Bravais lattices, classified into seven crystal systems (cubic, tetragonal, orthorhombic, monoclinic, triclinic, rhombohedral, and hexagonal). Each crystal system has one or more types of Bravais lattices associated with it that are defined by their rotational symmetry.

1. Cubic

A cubic crystal is characterized by a cubic unit cell, meaning that all three edges of the unit cell are of equal length, and the angles between them are 90 degrees. These relationships can be mathematically represented as:

a = b = c

α = β = γ = 90°

Cubic crystals exhibit a high degree of symmetry. They come in three types:

1. Simple Cubic (SC) Crystal: In a simple cubic crystal, also known as primitive cubic, each lattice point is located at the corners of the cube. SC structures are rare but can be found in certain elements, such as polonium.
2. Body-centered Cubic (BCC): In a body-centered cubic crystal, in addition to the lattice points at the corners of the cube, there is one lattice point located at the center of the cube. BCC structures can be observed in metals like iron, chromium, and tungsten.
3. Face-Centered Cubic (FCC) Crystal: In a face-centered cubic crystal, lattice points are located not only at the corners of the cube but also at the centers of each face of the cube. FCC structures are commonly found in aluminum, copper, and gold.

2. Tetragonal

A tetragonal crystal has a rectangular prism-shaped unit cell. It is characterized by three mutually perpendicular axes, two of which are of equal length, and the third is of a different length. The angles between the axes are all 90 degrees. Mathematically, the length and angle relationships are given as follows:

a = b ≠ c

α = β = γ = 90°

There are two types of tetragonal lattices:

1. Primitive Tetragonal Lattice: In this type, lattice points are located only at the corners of the unit cell, similar to a simple cubic lattice.
2. Body-Centered Tetragonal Lattice: In this type, lattice points are located at the corners and center of the unit cell. This is analogous to a body-centered cubic lattice but with unequal axis lengths.

Materials exhibiting a tetragonal lattice structure can be found in various substances, from metals like zirconium and titanium to minerals like rutile and cassiterite.

3. Orthorhombic

An orthorhombic crystal is characterized by three mutually perpendicular axes of different lengths. The angles between these axes are all 90 degrees, leading to a rectangular or parallelepiped-shaped unit cell.

Orthorhombic systems obey the following equations:

a ≠ b ≠ c

α = β = γ = 90°

There are four types of orthorhombic crystal structures:

1. Primitive Orthorhombic: In a primitive orthorhombic structure, also known as simple orthorhombic, the lattice points are located only at the corners of the unit cell.
2. Body-Centered Orthorhombic: In a body-centered orthorhombic structure, there is an additional lattice point at the center of the unit cell in addition to the lattice points at the corners.
3. Face-centered Orthorhombic: In a face-centered orthorhombic structure, lattice points are present not only at the corners of the unit cell but also at the centers of each face of the unit cell.
4. Base-Centered Orthorhombic: In this type of structure, in addition to lattice points at the corners of the unit cell, there is an additional lattice point at the center of one or more of the faces of the unit cell.

Orthorhombic crystals can be found in numerous mineral formations, such as the stunning topaz and the shimmering aragonite.

4. Rhombohedral

A rhombohedral crystal forms a structure with rhombus-shaped unit cells. It is characterized by its symmetry and non-orthogonal angles between its axes. The length and angle relationships are as follows:

a = b = c

α = β = γ ≠ 90°

5. Monoclinic

Monoclinic crystal is characterized by having three axes of different lengths, with two of them being perpendicular to each other, while the third axis is inclined relative to the other two. This results in a parallelogram-shaped unit cell. The angles between the axes are typically not right angles, with one angle being 90 degrees and the other two being unequal.

Bravais lattices having monoclinic systems obey the following relations:

a ≠ b ≠ c

β = γ = 90° and α ≠ 90°

There are two types of Bravais lattices in the monoclinic system:

1. Simple (Primitive) Monoclinic: This type of crystal system has one unique axis and unequal cell angles. The primitive monoclinic lattice has three unequal axes like the primitive orthorhombic lattice but with one axis inclined relative to the other two axes.
2. Base-centered Monoclinic: In this type, an additional lattice point is located at the center of one of the unit cells’ faces, creating a base-centered lattice.

Monoclinic crystals can be found in minerals such as gypsum, orthoclase feldspar, and clinopyroxene.

6. Triclinic

The triclinic lattice structure has three unequal axes and angles that are not right angles. It is the least symmetrical among all the crystal systems. There can be only one type of triclinic Bravais lattice, which is a primitive cell and obeys the following relationship:

a ≠ b ≠ c

α ≠ β ≠ γ ≠ 90°

Minerals like microcline, plagioclase, and labradorite exhibit triclinic symmetry, showcasing the diverse nature of crystalline structures found in nature.

7. Hexagonal

The hexagonal lattice consists of closely packed hexagonal layers stacked one on another. The crystal lattice is described by three equal axes intersecting at 120-degree angles in a plane, with a fourth axis perpendicular to the plane. A simple hexagonal cell has the following relations:

a = b ≠ c

α = β = 90° and γ = 120°

Graphite is an example of a crystal that crystallizes in the hexagonal crystal system

The following table summarizes the relationship among the lengths and angles of the above seven crystal systems.

Crystal System	Lengths	Angles
Cubic	a = b = c	α = β = γ = 90°
Tetragonal	a = b ≠ c	α = β = γ = 90°
Orthorhombic	a ≠ b ≠ c	α = β = γ = 90°
Rhombohedral	a = b = c	α = β = γ ≠ 90°
Monoclinic	a ≠ b ≠ c	β = γ = 90°; α ≠ 90°
Triclinic	a ≠ b ≠ c	α ≠ β ≠ γ ≠ 90°
Hexagonal	a = b ≠ c	α = β = 90°; γ = 120°