Bravais lattice

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

\mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}

where n_i are any integers and a_i 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.

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 of the lattice points.

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 at most 2 dimensions

In each of 0-dimensional and 1-dimensional space there is just one type of Bravais lattice.

In two dimensions, there are five Bravais lattices. They are oblique, rectangular, centered rectangular (rhombic), hexagonal, and square.^[2]

The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular (rhombic), 4 hexagonal, and 5 square

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
Triclinic
Monoclinic	P	C
Monoclinic
Orthorhombic	P	C	I	F
Orthorhombic
Tetragonal	P	I
Tetragonal
Rhombohedral	P
Rhombohedral
Hexagonal	P
Hexagonal
Cubic	P (pcc)	I (bcc)	F (fcc)
Cubic

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	$abc{\sqrt {1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}$
Monoclinic	$abc~\sin \alpha$
Orthorhombic	$abc$
Tetragonal	$a^{2}c$
Rhombohedral	$a^{3}{\sqrt {1-3\cos ^{2}\alpha +2\cos ^{3}\alpha }}$
Hexagonal	${\frac {3{\sqrt {3\,}}\,a^{2}c}{2}}$
Cubic	$a^{3}$

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

Bravais lattices in 4 dimensions

In four dimensions, there are 64 Bravais lattices. Of these, 23 are primitive and 41 are centered. Ten Bravais lattices split into enantiomorphic pairs.^[3]

References

^ Aroyo, Mois I. (2006). "Historical Introduction". International Tables for Crystallography. A1 (1.1). Springer: 2–5. doi:10.1107/97809553602060000537. Retrieved 2008-04-21. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ 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.
^ 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

External links

Smith, Walter Fox (2002). "The Bravais Lattices Song". {{cite web}}: Invalid |ref=harv (help)

[1] Aroyo, Mois I. (2006). "Historical Introduction". International Tables for Crystallography. A1 (1.1). Springer: 2–5. doi:10.1107/97809553602060000537. Retrieved 2008-04-21. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)

[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] 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

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