 Bravais lattice

Bravais lattice Definition

It is a distinct lattice that normally repeats in order to fill the whole space. In mathematics, a lattice is a regular, geometric arrangement of points, particles, or objects throughout an area or a space, especially the arrangement of ions or molecules in a crystalline solid. A Bravais lattice is generally any of the 14 possible three-dimensional configurations of points that describe the orderly arrangement of atoms in a crystal. It can be generated by three unit vectors, a1, a2 and a3 and a set of integers that are denoted as n1, n2 and n3. In this way, each lattice point is identified by a vector R to obtain:

R= n1a1+n2a2+n3a3

Where ni are any integers and ai are known as the primitive vectors which lie in various directions and extends the lattice.

In geometry and crystallography, this particular lattice is generally an infinite array of discrete points that are generated by another set of discrete translation operations as described in the above formula. A translation is overall defined as a rigid motion. It is also a function that moves every point a constant distance in a specified direction.

The discrete set of vectors must be closed under vector addition and subtraction. For any value of R, the lattice appears identical. Crystals have a periodic arrangement of one or more atoms that are repeated at each lattice point. A crystal lattice is formed when each point that represents one or more atoms are connected by lines. Due to this reason, the crystal when visualized from any of the lattice points appears the same.

Bravais lattice History

There are possibly 14 Bravias lattices that are mainly present in three-dimensional space. In 1850, Auguste Bravais, a French scientist showed that only these 14 forms of unit cells of the lattice entirely matched with the orderly arrangements of atoms in crystals.

Bravais lattice in two dimensions (2D)

Isometric, orthorhombic, monoclinic, tetragonal, triclinic, and hexagonal are the various forms of a lattice or crystal system. There are five types of lattices in two dimensions where the least symmetric Bravias lattice is the oblique lattice. If the angle between the two lattice vectors is 90, it gives rise to a higher symmetry of the cell that can either be a rectangle or square. However, it depends whether the unit cell vectors have same or different lengths. The rectangular lattice in turn can be classified into a primitive rectangular lattice and a centred rectangular lattice (rhombic) having an extra lattice point or atom at the centre. The lattice vectors with same length and forming an angle of 120give rise to a highly symmetrical hexagonal lattice.

Bravais lattice in three dimensions (3D)

There are 14 Bravias lattices in 3 dimensions obtained by combining one of the seven lattice or axial systems. Each Bravias lattice can be related to a distinct form. The different types of lattice centerings include:

• Base centered (A, B or C): Additional lattice point is present at the center of each of one pair of the cell faces.
• Primitive centering (P): Lattice points are present only on the cell corners.
• Face centered (F): In this type, one additional lattice point is located at the center of each of the faces of the cell.
• Body centered (I): One extra lattice point is present at the center of the cell.

Totally, there are exactly 7 x 6 = 42 combinations where most of them are identical. Monoclinic I lattice can be described by a monoclinic C lattice by taking different types of crystal axes. In a similar manner, all A- or B-centered lattices C can be demonstrated either by a C- or P- centering. Thus, the 14 Bravias lattices can be viewed from the table given below.

Volume of the unit cell is calculated by taking the cross product of a · b and c, where a · b is the dot product of a and b. The final product is a · b x c where a, b and c are considered as the lattice vectors. The volumes of the Bravias lattice are given below:

 Lattice system Volume Triclinic abc √ 1- cos2α- cos2β- cos2 γ + 2cosα cosβ cos γ Monoclinic abc sin α Orthorhombic abc Tetragonal a2c Rhombohedral a3√ 1- 3 cos2 α + 2 cos3 α Hexagonal 3√3 a2c/2 Cubic a3

Centered unit cells:

 Crystal System Possible            Variations Axial Distances     (edge lengths) Axial Angles Examples Cubic Primitive, Body centred, Face centred, End 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, End 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 (trigonal) Primitive a=b=c α = β = γ ≠ 90° Calcite (CaCO3, Cinnabar (HgS) Monoclinic Primitive, End 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

Bravais Lattice in four dimensions

There are 64 Bravais lattices in four dimensions out of which 23 are primitive and 41 are centered. 10 of these lattices are split into enantiomorphic pairs.

References

http://en.wikipedia.org/wiki/Bravais_lattice

http://users.aber.ac.uk/ruw/teach/334/bravais.php