Bravais Lattice
Bravais Lattice
In solid-state physics and crystallography, a Bravais lattice is a set of points constructed by translating a single point in discrete steps by a set of basis vectors. The Bravais lattice concept is fundamental in the description of crystal structures and helps in understanding the arrangement of atoms in a crystalline solid.
Definition
A Bravais lattice is an infinite array of discrete points with an arrangement and orientation that appear exactly the same from any of the points. The lattice can be generated by repeating a basic grouping of atoms, known as the basis, at each lattice point.
Fundamental Properties
- Translational Symmetry: A Bravais lattice is invariant under translation in three dimensions. This means that if you move from one lattice point to another, the surroundings look exactly the same.
- Discrete Points: The lattice points are separated by a finite distance in space. There are no intermediate points between lattice points that are part of the lattice.
- Three-Dimensional: Bravais lattices exist in three-dimensional space and are categorized by their geometrical properties.
The 14 Bravais Lattices
In three dimensions, there are 14 unique Bravais lattices that are grouped into 7 crystal systems. These lattices are distinguished by their cell parameters: the lengths of the cell edges (a, b, c) and the angles between them (α, β, γ).
| Crystal System | Lattice Type | Characteristics |
|---|---|---|
| Cubic | Simple (P) | a=b=c, α=β=γ=90° |
| Body-Centered (I) | a=b=c, α=β=γ=90° | |
| Face-Centered (F) | a=b=c, α=β=γ=90° | |
| Tetragonal | Simple (P) | a=b≠c, α=β=γ=90° |
| Body-Centered (I) | a=b≠c, α=β=γ=90° | |
| Orthorhombic | Simple (P) | a≠b≠c, α=β=γ=90° |
| Base-Centered (C) | a≠b≠c, α=β=γ=90° | |
| Body-Centered (I) | a≠b≠c, α=β=γ=90° | |
| Face-Centered (F) | a≠b≠c, α=β=γ=90° | |
| Hexagonal | Simple (P) | a=b≠c, α=β=90°, γ=120° |
| Rhombohedral | Simple (R) | a=b=c, α=β=γ≠90° |
| Monoclinic | Simple (P) | a≠b≠c, α=γ=90°, β≠90° |
| Base-Centered (C) | a≠b≠c, α=γ=90°, β≠90° | |
| Triclinic | Simple (P) | a≠b≠c, α≠β≠γ≠90° |
Lattice Points and Unit Cell
A lattice point is a position in space that has an environment that looks exactly the same in all directions. A unit cell is the smallest portion of a Bravais lattice that, when repeated through space, recreates the entire lattice without gaps or overlaps.
Unit Cell Parameters
- Lattice Constants: The lengths of the unit cell edges are known as lattice constants (a, b, c).
- Interaxial Angles: The angles between the edges of the unit cell are the interaxial angles (α, β, γ).
Volume of the Unit Cell
The volume of the unit cell (V) can be calculated using the lattice constants and the interaxial angles:
$$ V = abc\sqrt{1 - \cos^2(\alpha) - \cos^2(\beta) - \cos^2(\gamma) + 2\cos(\alpha)\cos(\beta)\cos(\gamma)} $$
For a cubic system where a = b = c and α = β = γ = 90°, the volume simplifies to:
$$ V = a^3 $$
Examples
Example 1: Simple Cubic Lattice
A simple cubic lattice has one lattice point per unit cell and a volume of $V = a^3$. The coordination number, which is the number of nearest neighbors to a lattice point, is 6.
Example 2: Body-Centered Cubic Lattice
A body-centered cubic (BCC) lattice has two lattice points per unit cell—one at each corner and one at the center of the cell. The volume is still $V = a^3$, but the coordination number is 8.
Example 3: Face-Centered Cubic Lattice
A face-centered cubic (FCC) lattice has four lattice points per unit cell—one at each corner and one at the center of each face. The volume is $V = a^3$, and the coordination number is 12.
Conclusion
Bravais lattices are essential for understanding the arrangement of atoms in crystalline solids. They provide a framework for classifying crystal structures and are a key concept in the study of materials science and solid-state physics. When preparing for exams, it is important to be familiar with the 14 Bravais lattices, their properties, and how they relate to the physical properties of materials.