Bravais lattice

Crystals are grouped into seven crystal systems, according to characteristic symmetry. The characteristic symmetry is the minimum symmetry of that system. The characteristic symmetry of a crystal is a combination of one or more rotations and inversions.

A lattice is a regular array of points. Each point must have the same number of neighbors as every other point and the neighbors must always be found at the same distances and directions. All points are in the same environment. A Bravais Lattice is a three dimensional lattice. A Bravais Lattice tiles space without any gaps or holes. There are 14 ways in which this can be accomplished. Lattices are characterized by translation symmetry.

Bravais Lattices contain seven crystal systems and four lattice centering types. The seven crystal systems along with their characteristic elements of symmetry are presented below.

No.TypeDescription
1PrimitiveLattice points on corners only. Symbol: P.
2Face CenteredLattice points on corners as well as centered on faces. Symbols: A (bc faces); B (ac faces); C (ab faces).
3All-Face CenteredLattice points on corners as well as in the centers of all faces. Symbol: F.
4Body-CenteredLattice points on corners as well as in the center of the unit cell body. Symbol: I.

The names below are linked to PDB files of Bravais Lattices. Each cell is easily created from simple geometry, and laws of sines and cosines. For clarity the centered points are colored differently from the other points, but in fact, are identical by translational symmetry.  For the hexagonal cell, the lattice is shown with and without the neighboring unit cells. The file containing the neighboring unit cells illustrates the hexagonal geometry. Three hexagonal unit cells form a hexagonal shape.

No.Bravais Lattice Type

Coordinate Description

Crystal System/
Characteristic Symmetry
1Primitive Cubic (P)

a = b = c

a = b = g = 90

 Cubic

Four 3-fold axes along a+b+c, –a+b+c, ab+c, –ab+c.

2Face Centered Cubic (F)

a = b = c

a = b = g = 90

3Body Centered Cubic (I)

a = b = c

a = b = g = 90

4Primitive Orthorhombic (P)

a =/= b =/= c

a = b = g = 90

 Orthorhombic

Three mutually perpendicular 2-fold rotation or rotatory-inversion axes along a, b, and c.

5Face Centered Orthorhombic (C)

a =/= b =/= c

a = b = g = 90

6Face Centered Orthorhombic (F)

a =/= b =/= c

a = b = g = 90

7Body Centered Orthorhombic (I)

a =/= b =/= c

a = b = g = 90

8Primitive Tetragonal (P)

a = b =/= c

a = b = g = 90

 Tetragonal

A single 4-fold rotation or rotatory-inversion axis along c.

9Body Centered Tetragonal (I)

a = b =/= c

a = b = g = 90

10Simple Monoclinic (P)

a =/= b =/= c

a = g = 90, b =/= 90

Monoclinic

A single 2-fold rotation or rotary inversion axis along b.

11B-Face Centered Monoclinic (C)

a =/= b =/= c

a = g = 90, b =/= 90

12Hexagonal (P)

one cell

three cells

a = b =/= c

a = b = 90, g = 120

AHexagonal

A single 6-fold rotation or rotatory-inversion axis along c.

13Triclinic (P)

a =/= b =/= c

a =/= b =/= g =/= 90

 Triclinic

Identity or inversion in any direction.

14Primative Rhombohedral (P)

a = b = c

a = b = g =/= 90

 Trigonal

Three-fold axis along c.

 

 

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Unit cell of SC, FCC and BCC - definition & explanation. Unit cell can be of primitive and non-primitive type....

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