Bravais lattice

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

where ni are any integers and ai are primitive vectors which lie in different directions (not necessarily mutually perpendicular) 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, or motif) 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.

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.

In 2 dimensions

2d-bravais
1 – oblique (monoclinic), 2 – rectangular (orthorhombic), 3 – centered rectangular (orthorhombic), 4 – hexagonal, and 5 – square (tetragonal).

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

Crystal family Point group
(Schönflies notation)
5 Bravais lattices
Primitive Centered
Monoclinic C2 Oblique
Orthorhombic D2 Rectangular Centered rectangular
Hexagonal D6 Hexagonal
Tetragonal D4 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 Area Axial distances (edge lengths) Axial angle
Monoclinic ab θ ≠ 90°
Orthorhombic ab θ = 90°
Hexagonal a = b θ = 120°
Tetragonal a = b θ = 90°

In 3 dimensions

Diamond lattice.stl
2×2×2 unit cells of a diamond cubic lattice

In three-dimensional space, there are 14 Bravais lattices. These are obtained by combining one of the seven lattice systems 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

Not all combinations of lattice systems 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 Schönflies 14 Bravais Lattices
Primitive (P) Base-centered (C) Body-centered (I) Face-centered (F)
Triclinic Ci Triclinic
Monoclinic C2h Monoclinic, simple Monoclinic, centered
Orthorhombic D2h Orthorhombic, simple Orthorhombic, base-centered Orthorhombic, body-centered Orthorhombic, face-centered
Tetragonal D4h Tetragonal, simple Tetragonal, body-centered
Hexagonal Rhombohedral D3d Rhombohedral
Hexagonal D6h Hexagonal
Cubic Oh 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 lattice systems are given below:

Crystal family Lattice system Volume Axial distances (edge lengths)[4] Axial angles[4] Corresponding examples
Triclinic (All remaining cases) K2Cr2O7, CuSO4·5H2O, H3BO3
Monoclinic ac α = γ = 90°, β ≠ 90° Monoclinic sulphur, Na2SO4·10H2O, PbCrO3
Orthorhombic abc α = β = γ = 90° Rhombic sulphur, KNO3, BaSO4
Tetragonal a = bc α = β = γ = 90° White tin, SnO2, TiO2, CaSO4
Hexagonal Rhombohedral a = b = c α = β = γ ≠ 90° Calcite (CaCO3), cinnabar (HgS)
Hexagonal a = b α = β = 90°, γ = 120° Graphite, ZnO, CdS
Cubic a = b = c α = β = γ = 90° NaCl, zinc blende, copper metal, KCl, Diamond, Silver

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.[5]

See also

References

  1. ^ Aroyo, Mois I.; Müller, Ulrich; Wondratschek, Hans (2006). "Historical Introduction". International Tables for Crystallography. A1 (1.1): 2–5. CiteSeerX 10.1.1.471.4170. doi:10.1107/97809553602060000537. Archived from the original on 2013-07-04. 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 978-0-471-11181-8. Retrieved 2008-04-21.
  3. ^ Based on the list of conventional cells found in Hahn (2002), p. 744
  4. ^ a b 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

External links

Born–von Karman boundary condition

Born–von Karman boundary conditions are periodic boundary conditions which impose the restriction that a wave function must be periodic on a certain Bravais lattice. Named after Max Born and Theodore von Kármán. This condition is often applied in solid state physics to model an ideal crystal. Born and von Karman published a series of articles in 1912 and 1913 that presented one of the first theories of specific heat of solids based on the crystaline hypothesis and included these boundary conditions.

The condition can be stated as

where i runs over the dimensions of the Bravais lattice, the ai are the primitive vectors of the lattice, and the Ni are integers (assuming the lattice has N cells where N=N1N2N3). This definition can be used to show that

for any lattice translation vector T such that:

Note, however, the Born–von Karman boundary conditions are useful when Ni are large (infinite).

The Born–von Karman boundary condition is important in solid state physics for analyzing many features of crystals, such as diffraction and the band gap. Modeling the potential of a crystal as a periodic function with the Born–von Karman boundary condition and plugging in Schrödinger's equation results in a proof of Bloch's theorem, which is particularly important in understanding the band structure of crystals.

Bragg plane

In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, , at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.

Considering the adjacent diagram, the arriving x-ray plane wave is defined by:

Where is the incident wave vector given by:

where is the wavelength of the incident photon. While the Bragg formulation assumes a unique choice of direct lattice planes and specular reflection of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as a source of secondary wavelets as described by the Huygens principle. Each scattered wave contributes to a new plane wave given by:

The condition for constructive interference in the direction is that the path difference between the photons is an integer multiple (m) of their wavelength. We know then that for constructive interference we have:

where . Multiplying the above by we formulate the condition in terms of the wave vectors, and :

Now consider that a crystal is an array of scattering centres, each at a point in the Bravais lattice. We can set one of the scattering centres as the origin of an array. Since the lattice points are displaced by the Bravais lattice vectors, , scattered waves interfere constructively when the above condition holds simultaneously for all values of which are Bravais lattice vectors, the condition then becomes:

An equivalent statement (see mathematical description of the reciprocal lattice) is to say that:

By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if is a vector of the reciprocal lattice. We notice that and have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector, , must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector, . This reciprocal space plane is the Bragg plane.

Brillouin zone

In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.

The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner-Seitz cell). Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice.

There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly n − 1 distinct Bragg planes.)

A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice (point group of the crystal).

The concept of a Brillouin zone was developed by Léon Brillouin (1889–1969), a French physicist.

Crystal structure

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.

The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive translation of the unit cell along its principal axes. The translation vectors define the nodes of the Bravais lattice.

The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called lattice parameters or cell parameters. The symmetry properties of the crystal are described by the concept of space groups. All possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups.

The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage, electronic band structure, and optical transparency.

Cubic crystal system

In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.

There are three main varieties of these crystals:

Primitive cubic (abbreviated cP and alternatively called simple cubic)

Body-centered cubic (abbreviated cI or bcc),

Face-centered cubic (abbreviated cF or fcc, and alternatively called cubic close-packed or ccp)Each is subdivided into other variants listed below. Note that although the unit cell in these crystals is conventionally taken to be a cube, the primitive unit cell often is not.

Fourier series

In mathematics, a Fourier series () is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the weighted sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials).

The study of Fourier series is a branch of Fourier analysis. The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series.

Hexagonal crystal family

In crystallography, the hexagonal crystal family is one of the 6 crystal families, which includes 2 crystal systems (hexagonal and trigonal) and 2 lattice systems (hexagonal and rhombohedral).

The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral.

Lattice plane

In crystallography, a lattice plane of a given Bravais lattice is a plane (or family of parallel planes) whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices) and intersect the Bravais lattice; equivalently, a lattice plane is any plane containing at least three noncollinear Bravais lattice points. All lattice planes can be described by a set of integer Miller indices, and vice versa (all integer Miller indices define lattice planes).

Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystals; this is known as a "cut-and-project" construction of a quasicrystal (and is typically also generalized to higher dimensions).

Monoclinic crystal system

In crystallography, the monoclinic crystal system is one of the 7 crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic system. They form a rectangular prism with a parallelogram as its base. Hence two vectors are perpendicular (meet at right angles), while the third vector meets the other two at an angle other than 90°.

Olivine

The mineral olivine ( ) is a magnesium iron silicate with the formula (Mg2+, Fe2+)2SiO4. Thus it is a type of nesosilicate or orthosilicate. It is a common mineral in Earth's subsurface but weathers quickly on the surface.

The ratio of magnesium to iron varies between the two endmembers of the solid solution series: forsterite (Mg-endmember: Mg2SiO4) and fayalite (Fe-endmember: Fe2SiO4). Compositions of olivine are commonly expressed as molar percentages of forsterite (Fo) and fayalite (Fa) (e.g., Fo70Fa30). Forsterite's melting temperature is unusually high at atmospheric pressure, almost 1,900 °C (3,450 °F), while fayalite's is much lower (about 1,200 °C [2,190 °F]). Melting temperature varies smoothly between the two endmembers, as do other properties. Olivine incorporates only minor amounts of elements other than oxygen, silicon, magnesium and iron. Manganese and nickel commonly are the additional elements present in highest concentrations.

Olivine gives its name to the group of minerals with a related structure (the olivine group)—which includes tephroite (Mn2SiO4), monticellite (CaMgSiO4) and kirschsteinite (CaFeSiO4).

Olivine's crystal structure incorporates aspects of the orthorhombic P Bravais lattice, which arise from each silica (SiO4) unit being joined by metal divalent cations with each oxygen in SiO4 bound to 3 metal ions. It has a spinel-like structure similar to magnetite but uses one quadrivalent and two divalent cations M22+ M4+O4 instead of two trivalent and one divalent cations.Olivine gemstones are called peridot and chrysolite.

Olivine rock is usually harder than surrounding rock and stands out as distinct ridges in the terrain. These ridges are often dry with little soil. Drought resistant scots pine is one of few trees that thrive on olivine rock. Olivine pine forest is unique to Norway. It is rare and found on dry olivine ridges in the fjord districts of Sunnmøre and Nordfjord. Olivine rock is hard and base-rich. The habitat is endangered by mining and road construction.

Orthorhombic crystal system

In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal.

Primitive cell

In geometry, crystallography, mineralogy, and solid state physics, a primitive cell is a minimum-volume cell (a unit cell) corresponding to a single lattice point of a structure with discrete translational symmetry. The concept is used particularly in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its primitive cell.

The primitive cell is a primitive unit. A primitive unit is a section of the tiling (usually a parallelogram or a set of neighboring tiles) that generates the whole tiling using only translations, and is as small as possible.

The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller.

Reciprocal lattice

In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, this first lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space (also known as momentum space or less commonly as K-space, due to the relationship between the Pontryagin duals momentum and position.) The reciprocal of a reciprocal lattice is the original direct lattice, since the two are Fourier transforms of each other.

The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron and X-ray diffraction, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal.

The Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice.

Rhombic enneacontahedron

A rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron.

Space group

In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space.

In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography (Hahn (2002)).

Tetragonal crystal system

In crystallography, the tetragonal crystal system is one of the 7 crystal systems. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base (a by a) and height (c, which is different from a).

Tetrahedron packing

In geometry, tetrahedron packing is the problem of arranging identical regular tetrahedra throughout three-dimensional space so as to fill the maximum possible fraction of space.

Currently, the best lower bound achieved on the optimal packing fraction of regular tetrahedra is 85.63%. Tetrahedra do not tile space, and an upper bound below 100% (namely, 1 − (2.6...)·10−25) has been reported.

Translation operator (quantum mechanics)

In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction.

More specifically, for any displacement vector , there is a corresponding translation operator that shifts particles and fields by the amount .

For example, if acts on a particle located at position , the result is a particle at position .

Translation operators are unitary.

Translation operators are closely related to the momentum operator; for example, a translation operator that moves by an infinitesimal amount in the direction has a simple relationship to the -component of the momentum operator. Because of this relationship, conservation of momentum holds when the translation operators commute with the Hamiltonian, i.e. when laws of physics are translation-invariant. This is an example of Noether's theorem.

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