Crystal Lattice

Crystal structure - Wikipedia, the free encyclopedia
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Crystal Lattice Structures
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crystalsIn mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A crystal structure is composed of a motif, a set of atoms arranged in a particular way, and a lattice. Motifs are located upon the points of a Bravais lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice constant. The symmetry properties of the crystal are embodied in its space group.A crystal's structure and symmetry play a role in determining many of its properties, such as cleavage (crystal), electronic band structure, and crystal optics.

==Unit cell==The crystal structure of a material or the arrangement of atoms in a crystal can be described in terms of its unit cell. The unit cell is a tiny box containing one or more motifs, a spatial arrangement of atoms. The units cells honeycomb (geometry) in three-dimensional space describes the bulk arrangement of atoms of the crystal. The unit cell is given by its Miller_index, the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (x_i, y_i, z_i) measured from a lattice point.

Although there are an infinite number of ways to specify a unit cell, for each crystal structure there is a conventional unit cell, which is chosen to display the full symmetry of the crystal (see below). However, the conventional unit cell is not always the smallest possible choice.A primitive unit cell of a particular crystal structure is the smallest possible volume one can construct with the arrangement of atoms in the crystal such that, when stacked, completely fills the space. This primitive unit cell does not always display all the symmetries inherent in the crystal. A Wigner-Seitz cell is a particular kind of primitive cell which has the same symmetry as the lattice.In an unit cell each atom has an identical environment when stacked in 3 dimensional space. In a primitive cell, each atom may not have the same environment.

There are only Crystal system that atoms can pack together to produce an infinite 3D space lattice in such a way that each lattice point has an identical environment to that around every other lattice point.

Classification of crystals by symmetry The defining property of a crystal is its inherent symmetry, by which we mean that under certain operations the crystal remains unchanged. For example, rotating the crystal 180 degrees about a certain axis may result in an atomic configuration which is identical to the original configuration. The crystal is then said to have a twofold rotational symmetry about this axis. In addition to rotational symmetries like this, a crystal may have symmetries in the form of mirror planes and translational symmetries, and also the so-called compound symmetries which are a combination of translation and rotation/mirror symmetries. A full classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified.

Crystal system {| align=right border=1 style=margin-left:1em|Crystal system|colspan=4 align=center| Lattices:|-|colspan=1 align=center| triclinic|-|rowspan=2 align=center| [monoclinic|| [image:Monoclinic-base-centered.svg|-|rowspan=2 align=center| orthorhombic|| [image:Orthorhombic-base-centered.svg|| image:Orthorhombic-body-centered.svg|| image:Orthorhombic-face-centered.svg|-|align=center| Hexagonal (crystal system)| image:Hexagonal lattice.svg|-|align=center| rhombohedral
(trigonal)]|-|rowspan=2 align=center| tetragonal|| [image:Tetragonal-body-centered.svg|-

|rowspan=2 align=center| Cubic (crystal system)
(isometric)|align=center| simple|align=center| body-centered|align=center| face-centered|-|| | | |}

The crystal systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each crystal system consists of a set of three axes in a particular geometrical arrangement. There are seven unique crystal systems. The simplest and most symmetric, the cubic (crystal system) (or isometric) system, has the symmetry of a Cube (geometry), that is, it exhibits four threefold rotational axes oriented at 109.5 degrees (the tetrahedral angle) with respect to each other. These threefold axes lie along the body diagonals of the cube. This definition of a cubic is correct, although many textbooks incorrectly state that a cube is defined by three mutually perpendicular axes of equal length – if this were true there would be far more than 14 Bravais lattices. The other six systems, in order of decreasing symmetry, are Hexagonal (crystal system), tetragonal, rhombohedral (also known as trigonal), orthorhombic, monoclinic and triclinic. Some crystallographers consider the hexagonal crystal system not to be its own crystal system, but instead a part of the trigonal crystal system. The crystal system and Bravais lattice of a crystal describe the (purely) translational symmetry of the crystal.

The Bravais lattices When the crystal systems are combined with the various possible lattice centerings, we arrive at the Bravais lattices. They describe the geometric arrangement of the lattice points, and thereby the translational symmetry of the crystal. In three dimensions, there are 14 unique Bravais lattices which are distinct from one another in the translational symmetry they contain.All crystalline materials recognized until now (not including quasicrystals) fit in one of these arrangements. The fourteen three-dimensional lattices, classified by crystal system, are shown to the right. The Bravais lattices are sometimes referred to as space lattices.

The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the 14 Bravais lattices. The characteristic rotation and mirror symmetries of the group of atoms, or unit cell, is described by its crystallographic point group.

Point and space groups The crystallographic point group or crystal class is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations can include reflection, which reflects the structure across a reflection plane, rotation, which rotates the structure a specified portion of a circle about a rotation axis, inversion which changes the sign of the coordinate of each point with respect to a center of symmetry or inversion point and improper rotation, which consists of a rotation about an axis followed by an inversion. Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.

The space group of the crystal structure is composed of the translational symmetry operations in addition to the operations of the point group. These include pure translations which move a point along a vector, screw axis, which rotate a point around an axis while translating parallel to the axis, and glide planes, which reflect a point through a plane while translating it parallel to the plane. There are 230 distinct space groups.

Physical properties Defects in crystals Real crystals feature crystallographic defect or irregularities in the ideal arrangements described above and it is these defects that critically determine many of the electrical and mechanical properties of real materials. In particular dislocations in the crystal lattice allow shear at much lower stress than that needed for a perfect crystal structure.

Crystal symmetry and physical properties Twenty of the 32 crystal classes are so-called piezoelectric, and crystals belonging to one of these classes (point groups) display piezoelectricity. All 20 piezoelectric classes lacka center of symmetry. Any material develops a dielectric polarizationwhen an electric field is applied, but a substance which has such anatural charge separation even in the absence of a field is called apolar material. Whether or not a material is polar is determinedsolely by its crystal structure. Only 10 of the 32 point groups are polar.All polar crystals are pyroelectric, so the 10 polar crystal classesare sometimes referred to as the pyroelectric classes.

There are a few crystal structures, notably the perovskite structure, which exhibit ferroelectric behaviour. This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarisation. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarised. This polarisation can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, it is important to note that, although they are called ferroelectrics, the effect is due to the crystal structure, not the presence of a ferrous metal.

Incommensurate crystals have period-varying translational symmetry. The period between nodes of symmetry is constant in most crystals. The distance between nodes in an incommensurate crystal is dependent on the number of nodes between it and the base node.

See also

Cleavage (crystal)
Crystal
Crystal engineering
Crystallography
Crystallographic point group
Crystallographic defect
Crystal growth
Liquid crystal
Miller Index
Patterson_function
Quasicrystals
Seed crystal

For more detailed information in specific technology applications see materials science, ceramic, or metallurgy.

External links

Appendix A from the manual for Atoms, software for XAFS
Intro to Minerals: Crystal Class and System
Introduction to Crystallography and Mineral Crystal Systems
Crystal planes and Miller indices
Interactive 3D Crystal models
Crystal Lattice Structures: Other Crystal Structure Web Sites

Crystal Atomic Lattice Viewer
Crystal Atomic Lattice Viewer. If your browser supports Java you should be able to rotate these structures using your mouse!!

Crystal structure - Wikipedia, the free encyclopedia
In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A crystal structure is composed of a motif, a set of atoms arranged in a ...

Crystal Lattice Structures
A list of the possible crystal lattice structures and coordinates of atoms in such crystals.

Crystal lattice viewer
Crystal lattice viewer Stuart Lynn (s.lynn@sms.ed.ac.uk), the School of Physics at the University of Edinburgh. Introduction The crystal lattice viewer is a visualization tool that ...

Crystal lattice viewer Documentation
The crystal lattice viewer Stuart Lynn (s.lynn@sms.ed.ac.uk), the School of Physics at the University of Edinburgh. Introduction The crystal lattice viewer is a visualization tool ...

Internet Microscope for Schools : Micrographs : Crystal Lattice
This is a high resolution image of the crystal lattice of an oxide ceramic (Ba 45 Nd 95 Ti 18 O 54). Its crystal lattice has a large unit cell (a=22.2Å, b=7.65Å, c=12.2Å), which ...

Explore Whipple Collections - Crystal lattice models
Crystal lattice models. Molecular model kits are designed to be re-used; models can be built and then taken apart again, but chemists often make permanent models of molecular ...

Amber Archive Jan 2004: AMBER: crystal lattice determination
From: Scott Brozell < sbrozell.scripps.edu > Date: Wed Jan 21 2004 - 16:51:37 GMT. Hello, Can one obtain crystal lattice information from an AMBER simulation

Amber Archive Jan 2004: Re: AMBER: crystal lattice determination
From: David A. Case < case.scripps.edu > Date: Fri Jan 23 2004 - 01:20:42 GMT. On Wed, Jan 21, 2004, Scott Brozell wrote: > > Can one obtain crystal lattice information from an ...

Glossary: Crystal lattice
The orderly, regular three-dimensional arrangement of atoms in a crystal. It can also refer to the structure of teeth and bone.