Bravais Lattice - Bravais Lattices in 3 Dimensions

Bravais Lattices in 3 Dimensions

The 14 Bravais lattices in 3 dimensions are obtained by coupling one of the 7 lattice systems (or axial systems) with one of the lattice centerings. Each Bravais lattice refers to a distinct lattice type.

The lattice centerings are:

  • Primitive (P): lattice points on the cell corners only.
  • Body (I): one additional lattice point at the center of the cell.
  • Face (F): one additional lattice point at center of each of the faces of the cell.
  • Base (A, B or C): one additional lattice point at the center of each of one pair of the cell faces.

Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total 7 × 6 = 42 combinations, but it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.

The 7 lattice systems The 14 Bravais lattices
Triclinic P
Monoclinic P C
Orthorhombic P C I F
Tetragonal P I
Rhombohedral
P
Hexagonal P
Cubic
P (pcc) I (bcc) F (fcc)


The volume of the unit cell can be calculated by evaluating a · b × c where a, b, and c are the lattice vectors. The volumes of the Bravais lattices are given below:

Lattice system Volume
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Rhombohedral
Hexagonal
Cubic


Centred Unit Cells :

Crystal System Possible Variations Axial Distances (edge lengths) Axial Angles Examples
Cubic Primitive, Body-centred, Face-centred a = b = c α = β = γ = 90° NaCl, Zinc Blende, Cu
Tetragonal Primitive, Body-centred a = b ≠ c α = β = γ = 90° White tin, SnO2, TiO2, CaSO4
Orthorhombic Primitive, Body-centred, Face-centred, Base-centred a ≠ b ≠ c α = β = γ = 90° Rhombic sulphur, KNO3, BaSO4
Hexagonal Primitive a = b ≠ c α = β = 90°, γ = 120° Graphite, ZnO, CdS
Rhombohedral Primitive a = b = c α = β = γ ≠ 90° Calcite (CaCO3, Cinnabar (HgS)
Monoclinic Primitive, Base-centred a ≠ b ≠ c α = γ = 90°, β ≠ 90° Monoclinic sulphur, Na2SO4.10H2O
Triclinic Primitive a ≠ b ≠ c α ≠ β ≠ γ ≠ 90° K2Cr2O7, CuSO4.5H2O, H3BO3

Read more about this topic:  Bravais Lattice

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