In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. Mineralogy is an Earth Science focused around the Chemistry, Crystal structure, and physical (including optical) properties of Minerals Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating A crystal structure is composed of a motif, a set of atoms arranged in a particular way, and a lattice. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny Motifs are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation 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 parameters. The Lattice Constant refers to the constant distance between Unit cells in a Crystal lattice. The symmetry properties of the crystal are embodied in its space group. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure A crystal's structure and symmetry play a role in determining many of its properties, such as cleavage, electronic band structure, and optical properties. Cleavage, in Mineralogy, is the tendency of crystalline materials to split along definite planes creating smooth surfaces of which there are several named types In Solid-state physics, the electronic band structure (or simply band structure) of a Solid describes ranges of Energy that an Electron Crystal optics is the branch of Optics that describes the behaviour of Light in Anisotropic media, that is media (such as Crystals
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Unit cell
The crystal structure of a material or the arrangement of atoms in a crystal structure 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. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The units cells stacked in three-dimensional space describes the bulk arrangement of atoms of the crystal. In Geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps The unit cell is given by its lattice parameters, 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 (xi,yi,zi) measured from a lattice point. Miller indices are a notation system in Crystallography for planes and directions in crystal (Bravais lattices In particular a family of Lattice planes
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. The Wigner-Seitz cell (named after E P Wigner and Frederick Seitz) is a geometrical construction which helps in the study of Crystalline material in In Geometry, Solid state physics and Mineralogy, particularly in describing Crystal structure, a primitive cell, is a minimum cell corresponding In a 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 seven possible crystal systems 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. A crystal system is a category of Space groups which characterize Symmetry of structures in three dimensions with Translational symmetry in three directions
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. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a A full classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified.
Crystal system
| The 7 Crystal systems (Defining Symmetry) |
The 14 Bravais Lattices: | |||
| triclinic (none) |
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| monoclinic (1 diad) |
simple | base-centered | ||
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| orthorhombic (3 perpendicular diads) |
simple | base-centered | body-centered | face-centered |
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| hexagonal (1 hexad) |
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| rhombohedral (1 triad) |
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| tetragonal (1 tetrad) |
simple | body-centered | ||
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| cubic (4 triads) |
simple | body-centered | face-centered | |
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The crystal systems are a grouping of crystal structures according to the axial system used to describe their lattice. In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching In Crystallography, the hexagonal is one of the 7 Crystal system, it contains 7 Point groups. In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic The cubic crystal system (or isometric) is a Crystal system where the Unit cell is in the shape of a Cube. A crystal system is a category of Space groups which characterize Symmetry of structures in three dimensions with Translational symmetry in three directions 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 (or isometric) system, has the symmetry of a cube, that is, it exhibits four threefold rotational axes oriented at 109. The cubic crystal system (or isometric) is a Crystal system where the Unit cell is in the shape of a Cube. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. 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. In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation The other six systems, in order of decreasing symmetry, are hexagonal, tetragonal, rhombohedral (also known as trigonal), orthorhombic, monoclinic and triclinic. In Crystallography, the hexagonal is one of the 7 Crystal system, it contains 7 Point groups. In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors 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.
Angles of the 7 basic crystalline structures
triclinic = 100 and 80 degrees, monoclinic = 100, 80 and 90 degrees, orthorhombic = 90 degrees, hexagonal = 90 and 120 degrees, rhombohedral = 100 and 80 degrees, tetragonal = 90 degrees, cubic = 90 degrees
The Bravais lattices
When the crystal systems are combined with the various possible lattice centerings, we arrive at the Bravais lattices. In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors. In Crystallography, the orthorhombic Crystal system is one of the seven Lattice Point groups Orthorhombic lattices result from stretching Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees In Crystallography, the rhombohedral (or trigonal) Crystal system is one of the seven lattice point groups named after the two-dimensional In Crystallography, the tetragonal Crystal system is one of the 7 lattice Point groups Tetragonal Crystal lattices result from stretching a cubic In Geometry and Crystallography, a Bravais lattice, named after Auguste Bravais, is an infinite set of points generated by a set of discrete translation 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. Quasicrystals are structural forms that are both ordered and nonperiodic 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. In Mineralogy and Crystallography, a crystal structure is a unique arrangement of Atoms in a Crystal. In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom
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. In Crystallography, a crystallographic point group is a set of Symmetry operations like rotations or reflections that leave a point fixed while moving each atom 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. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure These include pure translations which move a point along a vector, screw axes, 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 or impurities in crystals
Real crystals feature defects 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. Crystalline solids have a very regular atomic structure that is the local positions of atoms with respect to each other are repeated at the atomic scale When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue. [1] Impurities may also manifest as spin impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting ferromagnetic alloys may lead to different properties as first predicted in the late 1960s. Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it An alloy is a Solid solution or Homogeneous mixture of two or more elements, at least one of which is a Metal, which itself has [2][3]Dislocations in the crystal lattice allow shear at lower stress than that needed for a perfect crystal structure. In Materials science, a dislocation is a Crystallographic defect, or irregularity within a 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. Piezoelectricity is the ability of some materials (notably Crystals and certain Ceramics including bone to generate an Electric potential in response to Piezoelectricity is the ability of some materials (notably Crystals and certain Ceramics including bone to generate an Electric potential in response to All 21 piezoelectric classes lack a center of symmetry. Any material develops a dielectric polarization when an electric field is applied, but a substance which has such a natural charge separation even in the absence of a field is called a polar material. A dielectric is a nonconducting substance ie an insulator. The term was coined by William Whewell in response to a request from Michael Faraday. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes. Pyroelectricity is the ability of certain materials to generate an Electrical potential when they are heated or cooled
There are a few crystal structures, notably the perovskite structure, which exhibit ferroelectric behaviour. A perovskite is any material with the same type of Crystal structure as Calcium titanium oxide (CaTiO3 known as the perovskite structure Ferroelectricity is a physical property of a material whereby it exhibits a spontaneous electric polarization, the direction of which can be switched between equivalent This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarisation. Ferromagnetism is the basic mechanism by which certain materials (such as Iron) form Permanent magnets and/or exhibit strong interactions with Magnets it 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
- For more detailed information in specific technology applications see materials science, ceramic, or metallurgy. Materials Science or Materials Engineering is an interdisciplinary field involving the properties of matter and its applications to various areas of Science and The word ceramic is derived from the Greek word κεραμικός ( keramikos) Metallurgy is a domain of Materials science that studies the physical and chemical behavior of metallic elements, their intermetallic compounds, and their
References
- ^ Nikola Kallay (2000) Interfacial Dynamics, CRC Press, 741 pages ISBN:0824700066
- ^ C. Michael Hogan, (1969) Density of states of an insulating Ferromagnetic Alloy Physical Review 188, 870 - 874, [Issue 2 – December 1969
- ^ X. Y. Zhang and H. Suhl (1985) Phys. Rev. A 32, 2530 - 2533 (1985) [Issue 4 – October 1985
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
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