Bravais lattice

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in geometry and crystallography, a Bravais lattice, studied by Auguste Bravais (1850),[1] is an infinite array of discrete points in three dimensional space generated by a set of discrete translation operations described by:

where ni are any integers and ai are known as the primitive vectors which lie in different directions and span the lattice. This discrete set of vectors must be closed under vector addition and subtraction. For any choice of position vector R, the lattice looks exactly the same.

When the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its (finite) frontiers. A crystal is made up of a periodic arrangement of one or more atoms (the basis) repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, namely those separated by the translation of one unit cell (the motif).

Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups.

Bravais lattices in 2 dimensions[edit]

Further information: Lattice (group)
1 – oblique, 2 – rectangular, 3 – centered rectangular, 4 – hexagonal, and 5 – square.

In two-dimensional space, there are 5 Bravais lattices,[2] grouped into four crystal families.

Crystal family 5 Bravais lattices
Primitive Centered
Monoclinic Oblique
Orthorhombic Rectangular Centered rectangular
Hexagonal Hexagonal
Tetragonal Square

The unit cells are specified according to the relative lengths of the cell edges (a and b) and the angle between them (θ). The area of the unit cell can be calculated by evaluating the norm || a × b ||, where a and b are the lattice vectors. The properties of the crystal families are given below:

Crystal family Symmetry Area Axial distances (edge lengths) Axial angle
Monoclinic C2 ab θ ≠ 90°
Orthorhombic D2 ab θ = 90°
Hexagonal D6 a = b θ = 120°
Tetragonal D4 a = b θ = 90°

Bravais lattices in 3 dimensions[edit]

In three-dimensional space, there are 14 Bravais lattices. These are obtained by combining one of the six crystal families with one of the centering types. The centering types identify the locations of the lattice points in the unit cell as follows:

  • Primitive (P): lattice points on the cell corners only (sometimes called simple)
  • Base-centered (A, B, or C): lattice points on the cell corners with one additional point at the center of each face of one pair of parallel faces of the cell (sometimes called end-centered)
  • Body-centered (I): lattice points on the cell corners with one additional point at the center of the cell
  • Face-centered (F): lattice points on the cell corners with one additional point at the center of each of the faces of the cell
  • Rhombohedrally-centered (R): lattice points on the cell corners with two additional points along the longest body diagonal (only applies for the hexagonal crystal family)

Not all combinations of crystal families and centering types are needed to describe all of the possible lattices, as 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.[3]

Crystal family Lattice system 14 Bravais lattices
Primitive Base-centered Body-centered Face-centered Rhombohedrally-centered
triclinic Triclinic
monoclinic Monoclinic, simple Monoclinic, centered
orthorhombic Orthohombic, simple Orthohombic, base-centered Orthohombic, body-centered Orthohombic, face-centered
tetragonal Tetragonal, simple Tetragonal, body-centered
hexagonal rhombohedral Rhombohedral
hexagonal Hexagonal
cubic Cubic, simple Cubic, body-centered Cubic, face-centered

The unit cells are specified according to the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). The volume of the unit cell can be calculated by evaluating the triple product a · (b × c), where a, b, and c are the lattice vectors. The properties of the crystal families are given below:

Crystal family Lattice system Symmetry Volume Axial distances (edge lengths) Axial angles Corresponding examples
Triclinic Ci (All remaining cases)[4] K2Cr2O7, CuSO4·5H2O, H3BO3
Monoclinic C2h ac α = γ = 90°, β ≠ 90° Monoclinic sulfur, Na2SO4·10H2O
Orthorhombic D2h abc α = β = γ = 90° Rhombic sulfur, KNO3, BaSO4
Tetragonal D4h a = bc α = β = γ = 90° White tin, SnO2, TiO2, CaSO4
Hexagonal Rhombohedral D3d
(rhombohedral cell)
a = b = c
(rhombohedral cell)
α = β = γ ≠ 90°
(rhombohedral cell)
Calcite (CaCO3), cinnabar (HgS)
Hexagonal D6h a = b α = β = 90°, γ = 120° Graphite, ZnO, CdS
Cubic Oh a = b = c α = β = γ = 90° NaCl, zinc blende, copper metal

Bravais lattices in 4 dimensions[edit]

In four dimensions, there are 64 Bravais lattices. Of these, 23 are primitive and 41 are centered. Ten Bravais lattices split into enantiomorphic pairs.[5]

See also[edit]

References[edit]

  1. ^ Aroyo, Mois I.; Müller, Ulrich; Wondratschek, Hans (2006). "Historical Introduction". International Tables for Crystallography. Springer. A1 (1.1): 2–5. doi:10.1107/97809553602060000537. Retrieved 2008-04-21. 
  2. ^ Kittel, Charles (1996) [1953]. "Chapter 1". Introduction to Solid State Physics (Seventh ed.). New York: John Wiley & Sons. p. 10. ISBN 0-471-11181-3. Retrieved 2008-04-21. 
  3. ^ Based on the list of conventional cells found in Hahn (2002), p. 744
  4. ^ Hahn (2002), p. 758
  5. ^ Brown, Harold; Bülow, Rolf; Neubüser, Joachim; Wondratschek, Hans; Zassenhaus, Hans (1978), Crystallographic groups of four-dimensional space, New York: Wiley-Interscience [John Wiley & Sons], ISBN 978-0-471-03095-9, MR 0484179 

Further reading[edit]

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