Lattice Fundamentals
What is a Lattice?
A lattice is a mathematical concept describing a regular, infinite arrangement of points in space. In crystallography, these points represent the repeating positions where atoms, molecules, or groups of atoms (motifs) can be placed.
A lattice L is the set of all points that can be expressed as:
where:
- a, b, c are the lattice vectors (linearly independent)
- n₁, n₂, n₃ are integers
- r is any lattice point
Lattice vs Crystal Structure
It's crucial to distinguish between these concepts:
| Concept | Properties | |
|---|---|---|
| Lattice (Abstract) | Mathematical points in space | Defined by lattice vectors, Pure geometry, 14 Bravais lattices possible |
| Crystal Structure (Physical) | Atoms at specific positions | Lattice + motif/basis, Physical properties, Infinite structural variety |
Atoms are not lattice points. Atomic positions in a crystal are not necessarily lattice points - though they may coincide. Lattice points are mathematical abstractions that define the repeating pattern, while atoms occupy specific positions that may be offset from these points.
Mathematical Properties
Lattice Vector Matrix
The three lattice vectors form a lattice matrix:
Coordinate Transformations
The lattice matrix enables conversion between fractional coordinates (relative to lattice vectors) and Cartesian coordinates (absolute positions in space):
Example: For a cubic lattice with a = b = c = 2.5 Å and 90° angles:
A point at fractional coordinates (0.5, 0.5, 0.5) converts to:
Inverse Transformation
To convert from Cartesian to fractional coordinates, we use the inverse lattice matrix:
The inverse matrix M⁻¹ has reciprocal lattice vectors as its rows:
where V = det(M) is the unit cell volume.
Continuing the example: For the cubic case:
Converting the Cartesian point (1.0, 2.0, 1.5) Å back to fractional coordinates:
Unit Cell Volume
The volume of the unit cell (fundamental parallelepiped) is:
The Seven Lattice Systems
Constraints on the lattice vectors lead to seven distinct lattice systems, classified by their metric symmetry:
Bravais Lattices
Each lattice system can have different centering types, leading to the 14 Bravais lattices:
- Primitive (P): Lattice points only at corners
- Body-centered (I): Additional point at cell center
- Face-centered (F): Additional points at face centers
- Base-centered (A, B, C): Additional points at base centers
Not all combinations are crystallographically distinct, yielding exactly 14 unique Bravais lattices.
1. Triclinic
Constraints: None
Parameters:
Degrees of freedom: 6
Bravais lattices: 1
2. Monoclinic
Constraints:
Parameters:
Degrees of freedom: 4
Bravais lattices: 2
3. Orthorhombic
Constraints:
Parameters:
Degrees of freedom: 3
Bravais lattices: 4
4. Tetragonal
Constraints:
Parameters:
Degrees of freedom: 2
Bravais lattices: 2
5. Hexagonal
Constraints:
Parameters:
Degrees of freedom: 2
Bravais lattices: 1
6. Trigonal (Rhombohedral)
Constraints:
Parameters: All equal lengths and angles
Degrees of freedom: 2
Bravais lattices: 1
7. Cubic
Constraints:
Parameters: Single length parameter
Degrees of freedom: 1
Bravais lattices: 3
Interactive lattice
Now that you understand the fundamentals, explore how parameter constraints define different lattice systems using the full interactive tool:
a = 1.000, b = 1.000, c = 1.000
α = 90.0°, β = 90.0°, γ = 90.0°
Centering: P
Cell Volume: 1.000 ų