Bravais lattice

The Bravais lattice is a classification of the possible lattice systems (translation groups) in crystallography:

Integers and linearly independent vectors (in the case of three dimensions ) that span the grating ( primitive vectors). Bravais lattice are part of the classification of the area groups (and the mathematical derivation of the classification of the Bravais lattices can be found in the associated literature). The Bravais lattices are therefore a classification of possible translation groups regular grid of points.

In three dimensions there are fourteen Bravais lattices.

The presentation of the Bravais lattice is traditionally goes from the point groups and their classification into seven crystal systems (or 32 crystal classes, types of point groups ) from. Of the basic cell of the crystal system Bravais lattice caused by translation and be constructed by further addition of the grid points to the base cell. Here, the basic cell of the symmetry group of the crystal system is adjusted (see the discussion in the unit cell and the presentation below). For the Bravais lattice must still generally more grid points are added ( the chosen as a starting point the basic cell does not correspond to the primitive unit cell of the grid ). This can happen in six possible ways: face-centered - in respective opposite sides (A, B, C) or in any area (F ) - body-centered (I) and primitive (P, ie no addition of additional grid points ).

While the point group symmetry is visible in the outer crystal form and are made of the symmetry elements, rotation, reflection, inversion and rotation inversion the translations will be added for classification in the Bravais lattices in the crystal grid of the microscopic size ( Angstroms ), and not the external crystal form visible.

In this case, all grid points are considered equivalent. In the description of the crystal structure is generally to mathematical grid (defined by the possible translations ) are added or the description of the base, which can also consist of several atoms (crystal lattice structure is the same plus base ).

Auguste Bravais in 1849 classified the different possible translation lattice by in all directions he laid same parallelepiped cells together. The corners of the cells then produce a three-dimensional grid of points, (eg, atoms or molecules ) are the focal points of the crystal blocks in the real crystal.

In general, the forming is an oblique parallelepiped prism, in which all three side lengths and angles differ from one another. In this case it is a triclinic crystal system. Do the side lengths and / or angles other conditions, then higher symmetries may arise. The cubic crystal system requires, for example, right angles and equal length cell edges. Bravais noticed that there are lattice types that have a special feature: Your symmetry is higher than at the smallest possible cell would be readily recognizable. In halite, it is possible to select the area half the diagonal of a cube as the translation. However, the resulting grid has a rhombohedron with the angle of 70 ° 31 ' 44 " as the smallest parallelepiped. Reasons of symmetry, it is much more advisable to pick out a cube as a so-called unit cell of the lattice. This cubic unit cell is larger than the rhombohedral and contains the center of each face an additional grid point. This grid is called face-centered cubic.

• 2.2.1 Hexagonal crystal system
• 2.2.2 Trigonal crystal system
• 2.2.3 Monoclinic crystal system
• 2.2.4 triclinic crystal system

Use

The actually purely mathematical concept of the Bravais lattice often is used in the natural sciences, such as crystallography, mineralogy, materials science, solid state chemistry or solid state physics, as can so describe the arrangement of atoms within a crystal systematically. In this case, but not necessarily, each grid point is represented by an atom: the Bravais lattice supplies only the mathematical framework that (the base) is filled in a crystal structure of atoms. The crystal structure thus consists of the grating and of the base, which is repeated at each grid point, and is understood as a fundamental principle in crystallography. This results in, for example, the sodium chloride structure composed of a face-centered cubic lattice and a two-atom basis, according an Na cation and Cl - anion.

It has a special meaning in the structure determination of crystals. Based on the metric of the reflections in the reciprocal space and the systematic extinction integral the Bravais lattice of the crystal can be determined.

Classification

The Bravais lattices are identified by their point group assigned to the seven crystal systems. Corresponds to the reduced cell of the Bravais lattice the coordinate system of the crystal system, one speaks of a primitive lattice.

The further differentiation of the seven crystal systems of the 14 Bravais lattices is effected by arrangement of further grid points, either in the middle of the room ( space- centered or centered cubic ) to the center points of all the bounding surfaces ( face-centered ) or the centers of the two base surfaces (basic center) of the unit cell.

In the following, the Bravais lattices are after the crystal systems, with decreasing symmetry arranged.

Right angle (orthogonal ) axis systems

Cubic crystal system

• Highest symmetry
• Three equal axes at a 90 ° angle

Tetragonal crystal system

• Two axles of equal length, three 90 ° angles

Orthorhombic crystal system

• Also Rhombic crystal system
• Three 90 -degree angle, is not of long axes

Oblique axis systems

Hexagonal crystal system

• Two equally long axes in a plane angle of 120 °, the third axis perpendicular thereto

Trigonal crystal system

• Trigonal crystal structures can be also described in the hexagonal lattice:

• Hexagonal setting: a = b ≠ c, α = β = 90 °, γ = 120 ° (see figure above)

• As a special case, a rhombohedral centering occur:

• Three equal axes, three equal angles equal to 90 ° (see figure below)

• Not to be confused with the orthorhombic crystal system

Monoclinic crystal system

• Two 90 ° angles, is not of long axes

Triclinic crystal system

• Lowest symmetry of all grid
• No same angle, is not of long axes

Hermann- Mauguin symbols

In the Hermann- Mauguin symbols (see also point location ):

• Triclinic, P
• Monoclinic: primitive P2 / m, base - face-centered C2 / m
• Orthorhombic: Pmmm primitive, body-centered Immm, base face-centered Cmmm, face-centered Fmmm
• Hexagonal P6/mmm
• Rhombohedral, Rm
• Tetragonal: P4/mmm primitive, body-centered I4/mmm
• Kubisch: Pm3m primitive, body-centered Im3m, face-centered Fm3m

Bravais lattices in non- three-dimensional spaces

In two dimensions, there are five Bravais lattice, four primitive: the oblique lattice as well as four special types: the square, rectangular, hexagonal, and the centered - rectangular lattice, which is not primitive as a single. The surface of all types of three-dimensional grid consists of these two-dimensional lattice types. Therefore, they have a great importance in surface science and nanotechnology.

In the Four-dimensional there are 64 Bravais lattices ( 10 of which fall into enantiomorphic pairs, one of you this is not with it 54 ).