In

_{i}'', ''y_{i}'', ''z_{i}'') along the cell edges, measured from a reference point. It is only necessary to report the coordinates of a smallest asymmetric subset of particles. This group of particles may be chosen so that it occupies the smallest physical space, which means that not all particles need to be physically located inside the boundaries given by the lattice parameters. All other particles of the unit cell are generated by the symmetry operations that characterize the symmetry of the unit cell. The collection of symmetry operations of the unit cell is expressed formally as the
Image:Lattic_simple_cubic.svg, Simple cubic (P)
Image:Lattice_body_centered_cubic.svg, Body-centered cubic (I)
Image:Lattice_face_centered_cubic.svg, Face-centered cubic (F)

_{1}/''ℓ'', ''a''_{2}/''m'', and ''a''_{3}/''n'', or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it means that the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated so that it no longer contains that axis before its Miller indices are determined. The Miller indices for a plane are

_{2}. In the vast majority of _{2}.
Another example is elemental

The internal structure of crystals... Crystallography for beginners

Different types of crystal structure

Crystal planes and Miller indices

* Crystallography Open Database (with more than 140,000 crystal structures) {{Authority control Chemical properties

crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen drop", with its mean ...

, crystal structure is a description of the ordered arrangement of atom
An atom is the smallest unit of ordinary matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

s, ion
An ion () is a particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or ...

s or molecule
File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings.
A molecule is an electrically neutral group of two or more atoms held together by chemical bo ...

s 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
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...

in matter.
The smallest group of particles in the material that constitutes this repeating pattern is the unit cell
In crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen ...

of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive translation
Translation is the communication of the meaning
Meaning most commonly refers to:
* Meaning (linguistics), meaning which is communicated through the use of language
* Meaning (philosophy), definition, elements, and types of meaning discussed ...

of the unit cell along its principal axes. The translation vectors define the nodes of the Bravais lattice
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

.
The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constant
upright=1.3, Unit cell definition using parallelopiped with lengths ''a'', ''b'', ''c'' and angles between the sides given by ''α'', ''β'', ''γ''
The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a c ...

s, also called ''lattice parameters'' or ''cell parameters''. The symmetry
Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

properties of the crystal are described by the concept of space group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. All possible symmetric arrangements of particles in three-dimensional space may be described by the 230 space groups
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
The crystal structure and symmetry play a critical role in determining many physical properties, such as cleavage
Cleavage may refer to:
Science
* Cleavage (crystal), in mineralogy and materials science, a process of splitting a crystal
* Cleavage (geology), the foliation perpendicular to stress as a result of ductile deformation
* Cleavage (embryo), in embr ...

, electronic band structure
In solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state ph ...

, and optical transparency
In the field of optics, transparency (also called pellucidity or diaphaneity) is the physical property of allowing light to pass through the material without appreciable scattering of light. On a macroscopic scale (one where the dimensions invest ...

.
Unit cell

Crystal structure is described in terms of the geometry of arrangement of particles in the unit cell. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure. The geometry of the unit cell is defined as aparallelepiped
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

, providing six lattice parameters taken as the lengths of the cell edges (''a'', ''b'', ''c'') and the angles between them (α, β, γ). The positions of particles inside the unit cell are described by the fractional coordinates
In crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen ...

(''xspace group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the crystal structure.International Tables for Crystallography (2006). Volume A, Space-group symmetry.
Miller indices

Vectors and planes in a crystal lattice are described by the three-valueMiller index
Miller indices form a notation system in crystallography for planes in Bravais lattice, crystal (Bravais) lattices.
In particular, a family of lattice planes is determined by three integers ''h'', ''k'', and ''ℓ'', the ''Miller indices''. ...

notation. This syntax uses the indices ''ℓ'', ''m'', and ''n'' as directional parameters.Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
By definition, the syntax (''ℓmn'') denotes a plane that intercepts the three points ''a''integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

s with no common factors. Negative indices are indicated with horizontal bars, as in (13). In an orthogonal coordinate system for a cubic cell, the Miller indices of a plane are the Cartesian components of a vector normal to the plane.
Considering only (''ℓmn'') planes intersecting one or more lattice points (the ''lattice planes''), the distance ''d'' between adjacent lattice planes is related to the (shortest) reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is usually a periodic spatial fun ...

vector orthogonal to the planes by the formula
:$d\; =\; \backslash frac$
Planes and directions

The crystallographic directions are geometricline
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', a 2009 independent film by Nancy Schwartzman
Lite ...

s linking nodes (atom
An atom is the smallest unit of ordinary matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

s, ion
An ion () is a particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or ...

s or molecule
File:Pentacene on Ni(111) STM.jpg, A scanning tunneling microscopy image of pentacene molecules, which consist of linear chains of five carbon rings.
A molecule is an electrically neutral group of two or more atoms held together by chemical bo ...

s) of a crystal. Likewise, the crystallographic plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons''), a location in the multiverse
*Plane (Magic: Th ...

s are geometric ''planes'' linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:
*Optical propertiesThe optical properties of a material define how it interacts with light
Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that can be visual perception, perceived by the human eye. Visible l ...

: Refractive index
In optics, the refractive index (also known as refraction index or index of refraction) of a optical medium, material is a dimensionless number that describes how fast EM radiation, light travels through the material. It is defined as
:n = \frac, ...

is directly related to density (or periodic density fluctuations).
*Adsorption
of multilayer adsorption is a random distribution of molecules on the material surface.
Adsorption is the adhesion
Adhesion is the tendency of dissimilar Particle, particles or interface (matter), surfaces to cling to one another (Cohesion ...

and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes.
*Surface tension
Surface tension and hydrophobicity interact in this attempt to cut a water droplet.">water_droplet.html" ;"title="hydrophobicity interact in this attempt to cut a water droplet">hydrophobicity interact in this attempt to cut a water droplet.
...

: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.
*Microstructural defects
A defect is a physical, functional, or aesthetic attribute of a product or service that exhibits that the product or service failed to meet one of the desired specifications. Defect, defects or defected may also refer to:
Examples
* Angular defect ...

: Pores and crystallite
A crystallite is a small or even microscopic crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice ...

s tend to have straight grain boundaries following higher density planes.
*Cleavage
Cleavage may refer to:
Science
* Cleavage (crystal), in mineralogy and materials science, a process of splitting a crystal
* Cleavage (geology), the foliation perpendicular to stress as a result of ductile deformation
* Cleavage (embryo), in embr ...

: This typically occurs preferentially parallel to higher density planes.
*Plastic deformation
In engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encom ...

: Dislocation
In materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...

glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vectorIn materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials science s ...

) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice.
Some directions and planes are defined by symmetry of the crystal system. In monoclinic, rhombohedral, tetragonal, and trigonal/hexagonal systems there is one unique axis (sometimes called the principal axis) which has higher rotational symmetry
Rotational symmetry, also known as radial symmetry in biology, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it ...

than the other two axes. The basal plane is the plane perpendicular to the principal axis in these crystal systems. For triclinic, orthorhombic, and cubic crystal systems the axis designation is arbitrary and there is no principal axis.
Cubic structures

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted ''a''); similarly for the reciprocal lattice. So, in this common case, the Miller indices (''ℓmn'') and 'ℓmn''both simply denote normals/directions inCartesian coordinates
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft
Arts, entertainment and media
*Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

. For cubic crystals with lattice constant
upright=1.3, Unit cell definition using parallelopiped with lengths ''a'', ''b'', ''c'' and angles between the sides given by ''α'', ''β'', ''γ''
The lattice constant, or lattice parameter, refers to the physical dimension of unit cells in a c ...

''a'', the spacing ''d'' between adjacent (ℓmn) lattice planes is (from above):
:$d\_=\; \backslash frac$
Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:
*Coordinates in ''angle brackets'' such as denote a ''family'' of directions that are equivalent due to symmetry operations, such as 00 10 01or the negative of any of those directions.
*Coordinates in ''curly brackets'' or ''braces'' such as denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.
For face-centered cubic
200px, A network model of a primitive cubic system
In crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived fro ...

(fcc) and body-centered cubic
200px, A network model of a primitive cubic 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 cryst ...

(bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell
A supercell is a thunderstorm characterized by the presence of a mesocyclone: a deep, persistently rotating updraft
In meteorology, an updraft is a small-scale air current, current of rising air, often within a cloud.
Overview
Localized re ...

and hence are again simply the Cartesian directions.
Interplanar spacing

The spacing ''d'' between adjacent (''hkℓ'') lattice planes is given by: *Cubic: *:$\backslash frac\; =\; \backslash frac$ *Tetragonal: *:$\backslash frac\; =\; \backslash frac\; +\backslash frac$ *Hexagonal: *:$\backslash frac\; =\; \backslash frac\backslash left(\backslash frac\backslash right)+\backslash frac$ *Rhombohedral: *:$\backslash frac\; =\; \backslash frac$ *Orthorhombic: *:$\backslash frac\; =\; \backslash frac+\backslash frac+\backslash frac$ *Monoclinic: *:$\backslash frac\; =\backslash left(\backslash frac+\backslash frac+\backslash frac-\backslash frac\backslash right)\; \backslash csc^2\backslash beta$ *Triclinic: *:$\backslash frac\; =\; \backslash frac$Classification by symmetry

The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on the crystal lattice leaves it unchanged. All crystals havetranslational symmetry
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

in three directions, but some have other symmetry elements as well. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration; the crystal has twofold rotational symmetry about this axis. In addition to rotational symmetry, a crystal may have symmetry in the form of mirror planes, and also the so-called compound symmetries, which are a combination of translation and rotation or mirror symmetries. A full classification of a crystal is achieved when all inherent symmetries of the crystal are identified.
Lattice systems

Lattice systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each lattice system consists of a set of three axes in a particular geometric arrangement. All crystals fall into one of seven lattice systems. They are similar to, but not quite the same as the sevencrystal system
In crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen ...

s.
The simplest and most symmetric, the cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system
200px, A network ...

or isometric system, has the symmetry of a cube
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

, that is, it exhibits four threefold rotational axes oriented at 109.5° (the tetrahedral angle
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular Pyramid (geometry), pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex ( ...

) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon or 6-gon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regul ...

, tetragonal
In crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, froze ...

, rhombohedral
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

(often confused with the trigonal crystal system
In crystallography, the hexagonal crystal family is one of the six crystal family, crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigo ...

), orthorhombicIn crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic Lattice (group), lattices result from stretching a cubic crystal system, cubic lattice along two of its orthogonal pairs by two different factors, resul ...

, monoclinic
In crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, froze ...

and triclinic
Image:Microcline.jpeg, 200px, An example of the triclinic crystals, microcline
image:Triclinic.svg, 180px, Triclinic (a ≠ b ≠ c and α ≠ β ≠ γ )
In crystallography, the triclinic (or anorthic) crystal system is one of the 7 crystal syste ...

.
Bravais lattices

Bravais lattice
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

s, also referred to as ''space lattices'', describe the geometric arrangement of the lattice points, and therefore the translational symmetry of the crystal. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. All crystalline materials recognized today, not including quasicrystal
A quasiperiodic crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all dire ...

s, fit in one of these arrangements. The fourteen three-dimensional lattices, classified by lattice system, are shown above.
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 Bravais lattices. The characteristic rotation and mirror symmetries of the unit cell is described by its crystallographic point groupIn crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen dr ...

.
Crystal systems

A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.Point groups

Thecrystallographic point groupIn crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen dr ...

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 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''
*''Improper rotation
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

'', 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.
Space groups

In addition to the operations of the point group, thespace group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the crystal structure contains translational symmetry operations. 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.
*''Glide planes'', which reflect a point through a plane while translating it parallel to the plane.
There are 230 distinct space groups.
Atomic coordination

By considering the arrangement of atoms relative to each other, their coordination numbers (or number of nearest neighbors), interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.Close packing

The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer was placed directly over plane A, this would give rise to the following series: :...ABABABAB... This arrangement of atoms in a crystal structure is known as hexagonal close packing (hcp). If, however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises: :...ABCABCABC... This type of structural arrangement is known as cubic close packing (ccp). The unit cell of a ccp arrangement of atoms is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the planes of the fcc unit cell. There are four different orientations of the close-packed layers. The packing efficiency can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows: :$\backslash frac\; =\; \backslash frac\; =\; 0.7405...$ The 74% packing efficiency is the maximum density possible in unit cells constructed of spheres of only one size. Most crystalline forms of metallic elements are hcp, fcc, or bcc (body-centered cubic). Thecoordination numberIn chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central ion/m ...

of atoms in hcp and fcc structures is 12 and its atomic packing factorIn crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen dr ...

(APF) is the number mentioned above, 0.74. This can be compared to the APF of a bcc structure, which is 0.68.
Grain boundaries

Grain boundaries are interfaces where crystals of different orientations meet. Agrain boundary
of a polycrystalline
A crystallite is a small or even microscopic crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, for ...

is a single-phase interface, with crystals on each side of the boundary being identical except in orientation. The term "crystallite boundary" is sometimes, though rarely, used. Grain boundary areas contain those atoms that have been perturbed from their original lattice sites, dislocations
In materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...

, and impurities that have migrated to the lower energy grain boundary.
Treating a grain boundary geometrically as an interface of a single crystal
A single-crystal, or monocrystalline, solid is a material in which the crystal lattice of the entire sample is continuous and unbroken to the edges of the sample, with no grain boundaries.RIWD. "Reade Advanced Materials - Single Crystals". ''ww ...

cut into two parts, one of which is rotated, we see that there are five variables required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle of rotation of the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to this plane).
Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve strength, as described by the Hall–Petch
Grain-boundary strengthening (or Hall–Petch strengthening) is a method of strengthening materials by changing their average crystallite
A crystallite is a small or even microscopic crystal
A crystal or crystalline solid is a solid mater ...

relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the electrical
Electricity is the set of physical phenomena associated with the presence and motion
Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position
In physics, motion is the phenomenon in which an object changes its positio ...

and thermal conductivity
The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials ...

of the material. The high interfacial energy and relatively weak bonding in most grain boundaries often makes them preferred sites for the onset of corrosion and for the precipitation
In meteorology
Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant p ...

of new phases from the solid. They are also important to many of the mechanisms of creep.
Grain boundaries are in general only a few nanometers wide. In common materials, crystallites are large enough that grain boundaries account for a small fraction of the material. However, very small grain sizes are achievable. In nanocrystalline solids, grain boundaries become a significant volume fraction of the material, with profound effects on such properties as diffusion
File:DiffusionMicroMacro.gif, 250px, Diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diff ...

and plasticity
Plasticity may refer to:
Science
* Plasticity (physics), in engineering and physics, the propensity of a solid material to undergo permanent deformation under load
* Neuroplasticity, in neuroscience, how entire brain structures, and the brain its ...

. In the limit of small crystallites, as the volume fraction of grain boundaries approaches 100%, the material ceases to have any crystalline character, and thus becomes an amorphous solid
In condensed matter physics and materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellec ...

.
Defects and impurities

Real crystals featuredefects
A defect is a physical, functional, or aesthetic attribute of a product or service that exhibits that the product or service failed to meet one of the desired specifications. Defect, defects or defected may also refer to:
Examples
* Angular 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. 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. Impurities may also manifest as electron spin
The electron is a subatomic particle, symbol or , whose electric charge
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: ...

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
Ferromagnetism is the basic mechanism by which certain materials (such as iron
Iron () is a chemical element with Symbol (chemistry), symbol Fe (from la, Wikt:ferrum, ferrum) and atomic number 26. It is a metal that belongs to the first tra ...

alloy
An alloy is an admixture of metal
A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts Elec ...

s may lead to different properties as first predicted in the late 1960s. Dislocation
In materials science
The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...

s in the crystal lattice allow shear
Shear may refer to:
Textile production
* Animal shearing, the collection of wool from various species
**Sheep shearing
Sheep shearing is the process by which the woollen fleece of a sheep
Sheep (''Ovis aries'') are quadrupedal, ruminant m ...

at lower stress than that needed for a perfect crystal structure.
Prediction of structure

The difficulty of predicting stable crystal structures based on the knowledge of only the chemical composition has long been a stumbling block on the way to fully computational materials design. Now, with more powerful algorithms and high-performance computing, structures of medium complexity can be predicted using such approaches asevolutionary algorithms
In computational intelligence (CI), an evolutionary algorithm (EA) is a subset of evolutionary computation, a generic population-based metaheuristic optimization (mathematics), optimization algorithm. An EA uses mechanisms inspired by biological ev ...

, random sampling, or metadynamics.
The crystal structures of simple ionic solids (e.g., NaCl or table salt) have long been rationalized in terms of Pauling's rules
Pauling's rules are five rules published by Linus Pauling in 1929 for crystal structure prediction, predicting and rationalizing the crystal structures of ionic compounds.
First rule: the radius ratio rule
For typical ionic solids, the cations ...

, first set out in 1929 by Linus Pauling
Linus Carl Pauling (; February 28, 1901 – August 19, 1994) was an American chemist, biochemist, chemical engineer, peace activist, author, and educator. He published more than 1,200 papers and books, of which about 850 dealt with scientific top ...

, referred to by many since as the "father of the chemical bond". Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five d-orbitals in the transition metals are involved in bonding, with the remaining nonbonding d-orbitals being responsible for the magnetic properties. He, therefore, was able to correlate the number of d-orbitals in bond formation with the bond length as well as many of the physical properties of the substance. He subsequently introduced the metallic orbital, an extra orbital necessary to permit uninhibited resonance of valence bonds among various electronic structures.
In the resonating valence bond theoryIn condensed matter physics
Condensed matter physics is the field of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that st ...

, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": , , , , , etc. The choice of structure and the value of the axial ratio
Axial ratio, for any structure or shape with two or more axes, is the ratio
In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio o ...

(which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.
After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys, Hume-Rothery analyzed the trends in melting points, compressibilities and bond lengths as a function of group number in the periodic table in order to establish a system of valencies of the transition elements in the metallic state. This treatment thus emphasized the increasing bond strength as a function of group number. The operation of directional forces were emphasized in one article on the relation between bond hybrids and the metallic structures. The resulting correlation between electronic and crystalline structures is summarized by a single parameter, the weight of the d-electrons per hybridized metallic orbital. The "d-weight" calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between d-electrons and crystal structure thus becomes apparent.
In crystal structure predictions/simulations, the periodicity is usually applied, since the system is imagined as unlimited big in all directions. Starting from a triclinic structure with no further symmetry property assumed, the system may be driven to show some additional symmetry properties by applying Newton's Second Law on particles in the unit cell and a recently developed dynamical equation for the system period vectors
(lattice parameters including angles), even if the system is subject to external stress.
Polymorphism

Polymorphism is the occurrence of multiple crystalline forms of a material. It is found in many crystalline materials includingpolymer
A polymer (; Greek ''wikt:poly-, poly-'', "many" + ''wikt:-mer, -mer'', "part")
is a Chemical substance, substance or material consisting of very large molecules, or macromolecules, composed of many Repeat unit, repeating subunits. Due to their ...

s, mineral
In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. (20 ...

s, and metal
A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts Electrical resistivity and conductivity, el ...

s. According to Gibbs' rules of phase equilibria, these unique crystalline phases are dependent on intensive variables such as pressure and temperature. Polymorphism is related to allotropy
Allotropy or allotropism () is the property of some chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting o ...

, which refers to elemental solids. The complete morphology of a material is described by polymorphism and other variables such as crystal habit
In mineralogy
Mineralogy is a subject of geology specializing in the scientific study of the chemistry, crystal structure, and physical (including optical mineralogy, optical) properties of minerals and mineralized artifact (archaeology), artifa ...

, amorphous fraction or crystallographic defect
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids (see crystal structure). The word "crystallography" is derived from the Greek words ''crystallon'' "cold drop, frozen drop", with its mean ...

s. Polymorphs have different stabilities and may spontaneously and irreversibly transform from a metastable form (or thermodynamically unstable form) to the stable
A stable is a building in which livestock
Livestock is commonly defined as domesticated
Domestication is a sustained multi-generational relationship in which one group of organisms assumes a significant degree of influence over the reprod ...

form at a particular temperature. They also exhibit different , solubilities, and X-ray diffraction
X-ray crystallography (XRC) is the experimental science determining the atomic and molecular structure of a crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a ...

patterns.
One good example of this is the quartz
Quartz is a hard, crystalline mineral composed of silica (silicon dioxide). The atoms are linked in a continuous framework of SiO4 silicon-oxygen Tetrahedral molecular geometry, tetrahedra, with each oxygen being shared between two tetrahedra, ...

form of silicon dioxide
Silicon dioxide, also known as silica, is an oxide
of rutile. Ti(IV) centers are grey; oxygen centers are red. Notice that oxygen forms three bonds to titanium and titanium forms six bonds to oxygen.
An oxide () is a chemical compound that con ...

, or SiOsilicates
In chemistry
Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they unde ...

, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral units linked together by shared vertices in different arrangements. In different minerals the tetrahedra show different degrees of networking and polymerization. For example, they occur singly, joined together in pairs, in larger finite clusters including rings, in chains, double chains, sheets, and three-dimensional frameworks. The minerals are classified into groups based on these structures. In each of the 7 thermodynamically stable crystalline forms or polymorphs of crystalline quartz, only 2 out of 4 of each the edges of the tetrahedra are shared with others, yielding the net chemical formula for silica: SiOtin
Tin is a chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting only of atoms that all have the same numbers ...

(Sn), which is malleable near ambient temperatures but is brittle
tensile testpieces
A material is brittle if, when subjected to stress (physics), stress, it fracture
Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually ...

when cooled. This change in mechanical properties due to existence of its two major allotrope
Allotropy or allotropism () is the property of some chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting o ...

s, α- and β-tin. The two allotrope
Allotropy or allotropism () is the property of some chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting o ...

s that are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known as ''gray tin'' and ''white tin'' respectively. Two more allotropes, γ and σ, exist at temperatures above 161 °C and pressures above several GPa. White tin is metallic, and is the stable crystalline form at or above room temperature. Below 13.2 °C, tin exists in the gray form, which has a diamond cubic
The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, including α-tin, the se ...

crystal structure, similar to diamond
Diamond is a Allotropes of carbon, solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. At Standard conditions for temperature and pressure, room temperature and pressure, another solid form of c ...

, silicon
Silicon is a chemical element with the Symbol (chemistry), symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic lustre, and is a Tetravalence, tetravalent metalloid and semiconductor. It is a member ...

or germanium
Germanium is a chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting only of atoms that all have the same nu ...

. Gray tin has no metallic properties at all, is a dull gray powdery material, and has few uses, other than a few specialized semiconductor
A semiconductor material has an electrical conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that quantifies how strongly it resists electric curre ...

applications. Although the α–β transformation temperature of tin is nominally 13.2 °C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0 °C, and upon addition of Sb or Bi the transformation may not occur at all.
Physical properties

Twenty of the 32 crystal classes arepiezoelectric
Piezoelectricity (, ) is the electric charge that accumulates in certain solid materials—such as crystals, certain ceramics, and biological matter such as bone, DNA, and various proteins—in response to applied Stress (mechanics), mechanical st ...

, and crystals belonging to one of these classes (point groups) display piezoelectricity
Piezoelectricity is the electric charge
Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: ''positive'' and ''negative'' (comm ...

. All piezoelectric classes lack inversion symmetry
In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it ...

. Any material develops a dielectric
In electromagnetism
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ...

polarization when an electric field is applied, but a substance that has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
*Polar climate, the clim ...

. All polar crystals are pyroelectric
Pyroelectricity (from the two Greek words ''pyr'' meaning fire, and electricity) is a property of certain crystals which are naturally electrically polarized and as a result contain large electric fields. Pyroelectricity can be described as the ab ...

, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.
There are a few crystal structures, notably the perovskite structure
Image:Perovskite.jpg, Structure of a perovskite with general chemical formula ABX3. The red spheres are X atoms (usually oxygens), the blue spheres are B atoms (a smaller metal cation, such as Ti4+), and the green spheres are the A atoms (a larger m ...

, which exhibit ferroelectricFerroelectricity is a characteristic of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. All ferroelectrics are pyroelectric, with the additional property that t ...

behavior. This is analogous to ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials (such as iron) form permanent magnets, or are attracted to magnets. In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge o ...

, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarization. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, although they are called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).
See also

* * * * * * * * * * * * * *References

External links

The internal structure of crystals... Crystallography for beginners

Different types of crystal structure

Crystal planes and Miller indices

* Crystallography Open Database (with more than 140,000 crystal structures) {{Authority control Chemical properties

Condensed matter physics
Condensed matter physics
Condensed matter physics is the field of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that s ...

Crystallography
Materials science
Materials science includes those parts of chemistry, physics, geology and biology that deal with the physical, chemical or biological properties of materials. It is usually considered an applied science in which the properties it studies are used ...

Crystals
Conceptual systems