Editing Crystal structure

{{short description|Ordered arrangement of atoms, ions, or molecules in a crystalline material}}
[[File:Sodium-chloride-3D-ionic.png|thumb|Crystal structure of [[Sodium chloride|table salt]] (sodium in purple, chlorine in green)]]
In [[crystallography]], '''crystal structure''' is a description of ordered arrangement of [[atom]]s, [[ion]]s, or [[molecule]]s in a [[crystal|crystalline material]].<ref name="Solid State Physics 2010">{{cite book |title=Solid State Physics |edition=2nd |first1=J.R. |last1=Hook |first2=H.E. |last2=Hall |series=Manchester Physics Series |publisher=John Wiley & Sons |date=2010 |isbn=9780471928041 |url-access=registration |url=https://archive.org/details/solidstatephysic11cole_0 }}</ref> Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of [[Three-dimensional space (mathematics)|three-dimensional space]] in matter.

The smallest group of particles in a 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 (geometry)|translation]] of the unit cell along its principal axes. The translation vectors define the nodes of the [[Bravais lattice]].

The lengths of principal axes/edges, of the unit cell and angles between them are [[lattice constant]]s, also called ''lattice parameters'' or ''cell parameters''. The [[symmetry]] properties of a crystal are described by the concept of [[space groups]].<ref name="Solid State Physics 2010"/> All possible symmetric arrangements of particles in three-dimensional space may be described by 230 space groups.

The crystal structure and symmetry play a critical role in determining many physical properties, such as [[cleavage (crystal)|cleavage]], [[electronic band structure]], and [[crystal optics|optical transparency]].

== Unit cell ==
{{main|Unit cell}}
Crystal structure is described in terms of the geometry of the arrangement of particles in the unit cells. The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure.<ref>{{cite book|title=Basic Solid State Chemistry |first=Anthony R. |last=West |publisher=Wiley |edition=2nd |date=1999 |page=1 |isbn=978-0-471-98756-7}}</ref> The geometry of the unit cell is defined as a [[parallelepiped]], 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]] (''x<sub>i</sub>'', ''y<sub>i</sub>'', ''z<sub>i</sub>'') along the cell edges, measured from a reference point. It is thus only necessary to report the coordinates of a smallest asymmetric subset of particles, called the crystallographic asymmetric unit. The asymmetric unit 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 [[space group]] of the crystal structure.<ref name="Space-group symmetry">International Tables for Crystallography (2006). Volume A, Space-group symmetry.</ref>

<gallery class="center skin-invert-image">
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)
</gallery>

=== Miller indices ===
[[File:Miller Indices Cubes.svg|class=skin-invert-image|thumb|upright=1.2|Planes with different Miller indices in cubic crystals]]
Vectors and planes in a crystal lattice are described by the three-value [[Miller index]] notation. This syntax uses the indices ''h'', ''k'', and ''ℓ'' as directional parameters.<ref name="Physics 1991">Encyclopedia of Physics (2nd Edition), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</ref>

By definition, the syntax (''hkℓ'') denotes a plane that intercepts the three points ''a''<sub>1</sub>/''h'', ''a''<sub>2</sub>/''k'', and ''a''<sub>3</sub>/''ℓ'', 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, the planes do not intersect that axis (i.e., the intercept is "at infinity"). A plane containing a coordinate axis is translated to no longer contain that axis before its Miller indices are determined. The Miller indices for a plane are [[integer]]s with no common factors. Negative indices are indicated with horizontal bars, as in (1{{overbar|2}}3). 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 (''hkℓ'') planes intersecting one or more lattice points (the ''lattice planes''), the distance ''d'' between adjacent lattice planes is related to the (shortest) [[reciprocal lattice]] vector orthogonal to the planes by the formula
:<math>d = \frac{2\pi}  {|\mathbf{g}_{h k \ell}|}</math>

=== Planes and directions ===
The crystallographic directions are geometric [[line (mathematics)|line]]s linking nodes ([[atom]]s, [[ion]]s or [[molecule]]s) of a crystal. Likewise, the crystallographic [[plane (mathematics)|plane]]s are geometric ''planes'' linking nodes. Some directions and planes have a higher density of nodes. These high-density planes influence the behaviour of the crystal as follows:<ref name="Solid State Physics 2010"/>

*[[optics|Optical properties]]: [[Refractive index]] is directly related to density (or periodic density fluctuations).
*[[Adsorption]] and [[reactivity (chemistry)|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]]: 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.

[[File:Cristal densite surface.svg|class=skin-invert-image|thumb|Dense crystallographic planes]]

*Microstructural [[Crystallographic defect|defects]]: [[sintering|Pores]] and [[crystallite]]s tend to have straight grain boundaries following higher density planes.
*[[cleavage (crystal)|Cleavage]]: This typically occurs preferentially parallel to higher density planes.
*[[Plastic deformation]]: [[Dislocation]] glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation ([[Burgers vector]]) 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, trigonal, tetragonal, and hexagonal systems there is one unique axis (sometimes called the '''principal axis''') which has higher [[rotational symmetry]] 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 in [[Cartesian coordinates]]. For cubic crystals with [[lattice constant]] ''a'', the spacing ''d'' between adjacent (ℓmn) lattice planes is (from above):
:<math>d_{\ell mn}= \frac {a} { \sqrt{\ell ^2 + m^2 + n^2} }</math>
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 {{angbr|100}} denote a ''family'' of directions that are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.
*Coordinates in ''curly brackets'' or ''braces'' such as {100} 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]] (fcc) and [[body-centered cubic]] (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 (crystal)|supercell]] and hence are again simply the [[Cartesian coordinates|Cartesian directions]].

===Interplanar spacing===
The spacing '''''d''''' between adjacent (''hkℓ'') lattice planes is given by:<ref>{{cite web |title=4. Direct and reciprocal lattices |website=CSIC Dept de Cristalografia y Biologia Estructural |url= http://www.xtal.iqfr.csic.es/Cristalografia/parte_04-en.html |date=6 Apr 2017 |access-date=18 May 2017 }}</ref><ref>{{Cite book|last=Edington|first=J. W.|date=1975|title=Electron Diffraction in the Electron Microscope|language=en-gb|doi=10.1007/978-1-349-02595-4|isbn=978-0-333-18292-5}}</ref>

*Cubic:
*:<math>\frac {1} {d^{2}}= \frac {h^2+k^2+\ell^2} {a^2}</math>
*Tetragonal:
*:<math>\frac {1} {d^{2}}= \frac {h^2+k^2} {a^2}+\frac{\ell^2}{c^2}</math>
*Hexagonal:
*:<math>\frac {1} {d^{2}}= \frac{4}{3}\left(\frac{h^2+hk+k^2}{a^2}\right)+\frac{\ell^2}{c^2}</math>
*Rhombohedral ([[Unit cell|primitive setting]]):
*:<math>\frac {1} {d^{2}}= \frac{(h^2+k^2+\ell^2)\sin^2\alpha+2(hk+k\ell+h\ell)(\cos^2\alpha-\cos\alpha)}{a^2(1-3\cos^2\alpha+2\cos^3\alpha)}</math>
*Orthorhombic:
*:<math>\frac {1} {d^{2}}= \frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{\ell^2}{c^2}</math>
*Monoclinic:
*:<math>\frac {1} {d^{2}}=\left(\frac{h^2}{a^2}+\frac{k^2\sin^2\beta}{b^2}+\frac{\ell^2}{c^2}-\frac{2h\ell\cos\beta}{ac}\right) \csc^2\beta</math>
*Triclinic:
*:<math>\frac {1} {d^{2}}= \frac{\frac{h^2}{a^2}\sin^2\alpha+\frac{k^2}{b^2}\sin^2\beta+\frac{\ell^2}{c^2}\sin^2\gamma+\frac{2k\ell}{bc}(\cos\beta\cos\gamma-\cos\alpha)+\frac{2h\ell}{ac}(\cos\gamma\cos\alpha-\cos\beta)+\frac{2hk}{ab}(\cos\alpha\cos\beta-\cos\gamma)}{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma}</math>

== Classification by symmetry ==
{{main|Crystal system}}
The defining property of a crystal is its inherent symmetry. Performing certain symmetry operations on the crystal lattice leaves it unchanged. All crystals have [[translational symmetry]] 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.<ref>{{cite book|last1=Ashcroft|first1=N.|author-link1=Neil Ashcroft|last2=Mermin|first2=D.|author-link2=David Mermin|date=1976|title=Solid State Physics|publisher=Brooks/Cole (Thomson Learning, Inc.)|chapter=Chapter 7|isbn=978-0030493461}}</ref>

=== Lattice systems ===
Lattice systems are a grouping of crystal structures according to the point groups of their lattice. All crystals fall into one of seven lattice systems. They are related to, but not the same as the seven [[crystal system]]s. 
{| class="wikitable skin-invert-image"
|+Overview of common lattice systems
!scope="col" rowspan=2| Crystal family
!scope="col" rowspan=2| Lattice system
!scope="col" rowspan=2| Point group <br />([[Schoenflies notation|Schönflies notation]])
!scope="col" colspan=4| 14 Bravais lattices
|-
!scope="col"| Primitive (P)
!scope="col"| Base-centered (S)
!scope="col"| Body-centered (I)
!scope="col"| Face-centered (F)
|- align=center
!scope="row" colspan=2| [[Triclinic]] (a)
| C{{sub|i}}
| [[File:Triclinic.svg|80px|Triclinic]]
aP
|
|
|
|- align=center
!scope="row" colspan=2| [[Monoclinic]] (m)
| C{{sub|2h}}
| [[File:Monoclinic.svg|80px|Monoclinic, simple]]
mP
| [[File:Base-centered monoclinic.svg|80px|Monoclinic, centered]]
mS
|
|
|- align=center
!scope="row" colspan=2| [[Orthorhombic]] (o)
| D{{sub|2h}}
| [[File:Orthorhombic.svg|80px|Orthorhombic, simple]]
oP
| [[File:Orthorhombic-base-centered.svg|80px|Orthorhombic, base-centered]]
oS
| [[File:Orthorhombic-body-centered.svg|80px|Orthorhombic, body-centered]]
oI
| [[File:Orthorhombic-face-centered.svg|80px|Orthorhombic, face-centered]]
oF
|- align=center
!scope="row" colspan=2| [[Tetragonal]] (t)
| D{{sub|4h}}
| [[File:Tetragonal.svg|80px|Tetragonal, simple]]
tP
|
| [[File:Tetragonal-body-centered.svg|80px|Tetragonal, body-centered]]
tI
|
|- align=center
!scope="row" rowspan=2| [[Hexagonal crystal family|Hexagonal]] (h)
! Rhombohedral
| D{{sub|3d}}
| [[File:Rhombohedral.svg|80px|Rhombohedral]]
hR
|
|
|
|- align=center
!scope="row"| Hexagonal
| D{{sub|6h}}
| [[File:Hexagonal latticeFRONT.svg|80px|Hexagonal]]
hP
|
|
|
|- align=center
!scope="row" colspan=2| [[Cubic crystal system|Cubic]] (c)
| O{{sub|h}}
| [[File:Cubic.svg|80px|Cubic, simple]]
cP
|
| [[File:Cubic-body-centered.svg|80px|Cubic, body-centered]]
cI
| [[File:Cubic-face-centered.svg|80px|Cubic, face-centered]]
cF
|}
{{clear}}
The most symmetric, the [[cubic (crystal system)|cubic]] or isometric system, has the symmetry of a [[Cube (geometry)|cube]], that is, it exhibits four threefold rotational axes oriented at 109.5° (the [[tetrahedral angle]]) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are [[Hexagonal lattice system|hexagonal]], [[tetragonal]], [[rhombohedral lattice system|rhombohedral]] (often confused with the [[trigonal crystal system]]), [[orthorhombic]], [[monoclinic]] and [[triclinic]].

==== Bravais lattices ====
[[Bravais lattice]]s, also referred to as ''space lattices'', describe the geometric arrangement of the lattice points,<ref name="Physics 1991"/> 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]]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 group]].

===Crystal systems===
{{see also|Crystallographic point group#Isomorphisms}}

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.

{| class=wikitable
|+Overview of crystal systems
|-
!scope="col"| Crystal family
!scope="col"| Crystal system
!scope="col"| [[Point group]] / Crystal class
!scope="col"| [[Schönflies notation|Schönflies]]
!scope="col"| Point symmetry
!scope="col"| [[Symmetry number|Order]]
!scope="col"| [[Group theory#Abstract groups|Abstract group]]
|-
!scope="row" rowspan=2 colspan=2| [[triclinic crystal system|triclinic]]
| pedial	
| C{{sub|1}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 1
| trivial <math>\mathbb{Z}_1</math>
|-
| pinacoidal
| C{{sub|i}} (S{{sub|2}})
| [[centrosymmetric]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
!scope="row" rowspan=3 colspan=2 | [[monoclinic crystal system|monoclinic]]
| sphenoidal
| C{{sub|2}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| domatic
| C{{sub|s}} (C{{sub|1h}})
| [[Polar point group|polar]]
| 2
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_2</math>
|-
| [[prism (geometry)|prismatic]]
| C{{sub|2h}}	
| [[centrosymmetric]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=3 colspan=2| [[orthorhombic crystal system|orthorhombic]]
| rhombic-disphenoidal
| D{{sub|2}} (V)
| [[Chirality (chemistry)|enantiomorphic]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| rhombic-[[Pyramid (geometry)|pyramidal]]
| C{{sub|2v}}	
| [[Polar point group|polar]]
| 4
| [[Klein four-group|Klein four]] <math>\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2</math>
|-
| rhombic-[[dipyramid]]al
| D{{sub|2h}} (V{{sub|h}})	
| [[centrosymmetric]]
| 8
| <math>\mathbb{V}\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=7 colspan=2| [[tetragonal crystal system|tetragonal]]
| tetragonal-pyramidal	
| C{{sub|4}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-disphenoidal
| S{{sub|4}}	
| [[non-centrosymmetric]]
| 4
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_4</math>
|-
| tetragonal-dipyramidal
| C{{sub|4h}}	
| [[centrosymmetric]]
| 8
| <math>\mathbb{Z}_4\times\mathbb{Z}_2</math>
|-
| tetragonal-trapezohedral
| D{{sub|4}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-pyramidal
| C{{sub|4v}}	
| [[Polar point group|polar]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| tetragonal-scalenohedral
| D{{sub|2d}} (V{{sub|d}})
| [[non-centrosymmetric]]
| 8
| [[Dihedral group|dihedral]] <math>\mathbb{D}_8 = \mathbb{Z}_4\rtimes\mathbb{Z}_2</math>
|-
| ditetragonal-dipyramidal
| D{{sub|4h}}	
| [[centrosymmetric]]
| 16
| <math>\mathbb{D}_8\times\mathbb{Z}_2</math>
|-
!scope="row" rowspan=12|[[hexagonal crystal family|hexagonal]] !! rowspan=5 | trigonal 
| trigonal-pyramidal	
| C{{sub|3}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 3
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_3</math>
|-
| rhombohedral
| C{{sub|3i}} (S{{sub|6}})
| [[centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-trapezohedral
| D{{sub|3}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-pyramidal
| C{{sub|3v}}	
| [[Polar point group|polar]]
| 6
| [[Dihedral group|dihedral]] <math>\mathbb{D}_6 = \mathbb{Z}_3\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-scalenohedral
| D{{sub|3d}}	
| [[centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
!scope="row" rowspan=7 | hexagonal
| hexagonal-pyramidal	
| C{{sub|6}}
| [[Chirality (chemistry)|enantiomorphic]] [[Polar point group|polar]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| trigonal-dipyramidal
| C{{sub|3h}}
| [[non-centrosymmetric]]
| 6
| [[Cyclic group|cyclic]] <math>\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2</math>
|-
| hexagonal-dipyramidal
| C{{sub|6h}}	
| [[centrosymmetric]]
| 12
| <math>\mathbb{Z}_6\times\mathbb{Z}_2</math>
|-
| hexagonal-trapezohedral
| D{{sub|6}}	
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-pyramidal
| C{{sub|6v}}	
| [[Polar point group|polar]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| ditrigonal-dipyramidal
| D{{sub|3h}}	
| [[non-centrosymmetric]]
| 12
| [[Dihedral group|dihedral]] <math>\mathbb{D}_{12} = \mathbb{Z}_6\rtimes\mathbb{Z}_2</math>
|-
| dihexagonal-dipyramidal
| D{{sub|6h}}	
| [[centrosymmetric]]
| 24
| <math>\mathbb{D}_{12}\times\mathbb{Z}_2</math> 
|-
!scope="row" rowspan=5 colspan=2 | [[cubic crystal system|cubic]]
| tetartoidal
| T 
| [[Chirality (chemistry)|enantiomorphic]]
| 12
| [[alternating group|alternating]] <math>\mathbb{A}_4</math>
|-
| diploidal
| T{{sub|h}}	
| [[centrosymmetric]]
| 24
| <math>\mathbb{A}_4\times\mathbb{Z}_2</math> 
|-
| gyroidal
| O
| [[Chirality (chemistry)|enantiomorphic]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| hextetrahedral
| T{{sub|d}}	
| [[non-centrosymmetric]]
| 24
| [[symmetric group|symmetric]] <math>\mathbb{S}_4</math>
|-
| hexoctahedral
| O{{sub|h}}	
| [[centrosymmetric]]
| 48
| <math>\mathbb{S}_4\times\mathbb{Z}_2</math> 
|}
{{clear}}
In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

==== Point 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 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]]'', 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, the [[space group]] 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.<ref name=Sands>
{{cite book |title=Introduction to Crystallography |author=Donald E. Sands |chapter-url=https://books.google.com/books?id=h_A5u5sczJoC&pg=PA71 |chapter=§4-2 Screw axes and §4-3 Glide planes |pages= 70–71 |isbn=978-0486678399 |publisher=Courier-Dover |year=1994 |edition=Reprint of WA Benjamin corrected 1975}} 
</ref>
*''Glide planes'', which reflect a point through a plane while translating it parallel to the plane.<ref name=Sands/>

There are 230 distinct space groups.

== Atomic coordination ==
By considering the arrangement of atoms relative to each other, their coordination numbers, interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.<ref name="Physics 1994">{{cite book |title=McGraw Hill Encyclopaedia of Physics |edition=2nd |editor1-first=C.B. |editor1-last=Parker |date=1994 |publisher=McGraw-Hill |isbn=978-0070514003 |url-access=registration |url=https://archive.org/details/mcgrawhillencycl1993park }}</ref>

=== Close packing ===
{{Main|Close-packing of equal spheres}}
[[File:close packing.svg|thumb|upright=1.3|The hpc lattice (left) and the ccf lattice (right)]]
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 were 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 {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers.

===APF and CN===
{{main|Atomic packing factor|Coordination number}}
One important characteristic of a crystalline structure is its atomic packing factor (APF). This is calculated by assuming that all the atoms are identical spheres, with a radius large enough that each sphere abuts on the next. The atomic packing factor is the proportion of space filled by these spheres which can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows:
:<math>\mathrm{APF} = \frac{N_\mathrm{particle} V_\mathrm{particle}}{V_\text{unit cell}}</math>

Another important characteristic of a crystalline structure is its coordination number (CN). This is the number of nearest neighbours of a central atom in the structure.

The APFs and CNs of the most common crystal structures are shown below:

{| class=wikitable
! Crystal structure
! Atomic packing factor
! Coordination number<br/>([[Coordination geometry|Geometry]])
|-
! [[Diamond cubic]]
| 0.34
| 4 ([[Tetrahedron]])
|-
! [[Simple cubic]]
| 0.52<ref name=Ellis>{{cite book|first1=Arthur B.|display-authors=etal|last1=Ellis|title=Teaching General Chemistry: A Materials Science Companion|date=1995|publisher=American Chemical Society|location=Washington, DC|isbn=084122725X|edition=3rd}}</ref>
| 6 ([[Octahedron]])
|-
! [[Body-centered cubic]] (BCC)
| 0.68<ref name=Ellis/>
| 8 ([[Cube]])
|-
! [[Face-centered cubic]] (FCC)
| 0.74<ref name=Ellis/>
| 12 ([[Cuboctahedron]])
|-
! [[Hexagonal crystal system|Hexagonal close-packed]] (HCP)
| 0.74<ref name=Ellis/>
| 12 ([[Triangular orthobicupola]])
|}

The 74% packing efficiency of the FCC and HCP is the maximum density possible in unit cells constructed of spheres of only one size.

===Interstitial sites===
{{Expand section|date=August 2022}}
{{main|Interstitial site}}
[[Image:Sites interstitiels cubique a faces centrees.svg|thumb|200px|''Octahedral'' (red) and ''tetrahedral'' (blue) interstitial sites in a [[face-centered cubic]] lattice.]]
Interstitial sites refer to the empty spaces in between the atoms in the crystal lattice. These spaces can be filled by oppositely charged ions to form multi-element structures. They can also be filled by impurity atoms or self-interstitials to form [[interstitial defect]]s.
{{clear}}

== Defects and impurities ==
{{main|Crystallographic defect}}
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.

=== Impurities ===
{{Expand section|date=August 2022}}
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.<ref>{{cite book|first=Nikola |last=Kallay |date=2000 |url=https://books.google.com/books?id=ZXsBk20WO1sC |title=Interfacial Dynamics |publisher=CRC Press |isbn=978-0824700065}}</ref> Impurities may also manifest as [[electron 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]] [[alloy]]s may lead to different properties as first predicted in the late 1960s.<ref>{{cite journal|doi = 10.1103/PhysRev.188.870|title = Density of States of an Insulating Ferromagnetic Alloy|year = 1969|author = Hogan, C. M.|journal = Physical Review|volume = 188|issue = 2|pages = 870–874|bibcode = 1969PhRv..188..870H }}</ref><ref>{{cite journal|doi = 10.1103/PhysRevA.32.2530|title = Spin-wave-related period doublings and chaos under transverse pumping|year = 1985|author = Zhang, X. Y.|journal = Physical Review A|volume = 32|pages = 2530–2533|pmid = 9896377|first2 = H|issue = 4|last2 = Suhl|bibcode = 1985PhRvA..32.2530Z }}</ref>

=== Dislocations ===
{{Expand section|date=August 2022}}
{{main|Dislocation}}
Dislocations in a crystal lattice are line defects that are associated with local stress fields. Dislocations allow [[Shear stress|shear]] at lower stress than that needed for a perfect crystal structure.<ref>{{cite book |title= Mechanical Behavior of Materials |last= Courtney |first= Thomas |year= 2000 |publisher= Waveland Press |location= Long Grove, IL |isbn= 978-1-57766-425-3 |pages= 85}}</ref> The local stress fields result in interactions between the dislocations which then result in strain hardening or [[cold working]].

=== Grain boundaries ===
{{Main|Grain boundary}}
Grain boundaries are interfaces where crystals of different orientations meet.<ref name="Physics 1991"/> A grain boundary 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]], and impurities that have migrated to the lower energy grain boundary.

Treating a grain boundary geometrically as an interface of a [[single crystal]] 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).<ref name="Physics 1994"/>

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]] relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the [[electrical conductivity|electrical]] and [[thermal conductivity]] 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 (chemistry)|precipitation]] of new phases from the solid. They are also important to many of the mechanisms of [[creep (deformation)|creep]].<ref name="Physics 1994"/>

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]] and [[plasticity (physics)|plasticity]]. 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]].<ref name="Physics 1994"/>

== Prediction of structure ==
{{Main| Crystal structure prediction}}
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 as [[evolutionary algorithms]], 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]], first set out in 1929 by [[Linus Pauling]], referred to by many since as the "father of the chemical bond".<ref name="Paulingsrules">{{cite journal|author= L. Pauling|year = 1929|title = The principles determining the structure of complex ionic crystals|journal = [[J. Am. Chem. Soc.]]|volume = 51|issue = 4|pages= 1010–1026|doi= 10.1021/ja01379a006| bibcode=1929JAChS..51.1010P |author-link = Linus Pauling}}</ref> 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. Pauling was therefore able to correlate the number of d-orbitals in bond formation with the bond length, as well as with 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.<ref>{{cite journal|doi=10.1103/PhysRev.54.899|title=The Nature of the Interatomic Forces in Metals|year=1938|last1=Pauling|first1=Linus|journal=Physical Review|volume=54|issue=11|pages=899–904|bibcode = 1938PhRv...54..899P }}</ref>

In the [[resonating valence bond theory]], 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": {{frac|1|2}}, {{frac|1|3}}, {{frac|2|3}}, {{frac|1|4}}, {{frac|3|4}}, etc. The choice of structure and the value of the [[axial ratio]] (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.<ref>{{cite journal|doi=10.1021/ja01195a024|year=1947|last1=Pauling|first1=Linus|title=Atomic Radii and Interatomic Distances in Metals|journal=Journal of the American Chemical Society|volume=69|issue=3|pages=542–553|bibcode=1947JAChS..69..542P }}</ref><ref>{{cite journal|doi=10.1098/rspa.1949.0032|title=A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds|year=1949|last1=Pauling|first1=L.|journal=[[Proceedings of the Royal Society A]]|volume=196|issue=1046|pages=343–362|bibcode = 1949RSPSA.196..343P |doi-access=free}}</ref>

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.<ref>{{cite journal|doi=10.1098/rspa.1951.0172|title=The Valencies of the Transition Elements in the Metallic State|year=1951|last1=Hume-rothery|first1=W.|first2=H. M.|first3=R. J. P.|journal=[[Proceedings of the Royal Society A]]|volume=208|issue=1095|pages=431|last2=Irving|last3=Williams|bibcode = 1951RSPSA.208..431H |s2cid=95981632}}</ref> 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.<ref>{{cite journal|doi=10.1098/rspa.1957.0073|title=On the Relation between Bond Hybrids and the Metallic Structures|year=1957|last1=Altmann|first1=S. L.|first2=C. A.|first3=W.|journal=[[Proceedings of the Royal Society A]]| volume=240| issue=1221| pages=145| last2=Coulson| last3=Hume-Rothery| bibcode = 1957RSPSA.240..145A |s2cid=94113118}}</ref>

In crystal structure predictions/simulations, the periodicity is usually applied, since the system is imagined as being unlimited 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
<ref>{{cite journal|doi=10.1139/cjp-2014-0518|title=Dynamical equations for the period vectors in a periodic system under constant external stress|year=2015|last1=Liu|first1=Gang|journal=[[Can. J. Phys.]]| volume=93| issue=9| pages=974–978|bibcode = 2015CaJPh..93..974L |arxiv=cond-mat/0209372|s2cid=54966950}}</ref> (lattice parameters including angles), even if the system is subject to external stress.

== Polymorphism ==
[[File:Quartz, Tibet.jpg|thumb|[[Quartz]] is one of the several [[crystalline]] forms of [[silica]], SiO<sub>2</sub>. The most important forms of silica include: [[quartz|α-quartz]], [[quartz|β-quartz]], [[tridymite]], [[cristobalite]], [[coesite]], and [[stishovite]].]]
[[Polymorphism (materials science)|Polymorphism]] is the occurrence of multiple crystalline forms of a material. It is found in many crystalline materials including [[polymer]]s, [[mineral]]s, and [[metal]]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]], which refers to [[chemical elements|elemental solids]]. The complete morphology of a material is described by polymorphism and other variables such as [[crystal habit]], [[Amorphous solid|amorphous fraction]] or [[crystallographic defect]]s. Polymorphs have different stabilities and may spontaneously and irreversibly transform from a metastable form (or thermodynamically unstable form) to the [[Chemical stability|stable]] form at a particular temperature.<ref>{{cite journal | doi=10.1007/s10853-010-5113-0 | title=Review of the anatase to rutile phase transformation | date=2011 | last1=Hanaor | first1=Dorian A. H. | last2=Sorrell | first2=Charles C. | journal=Journal of Materials Science | volume=46 | issue=4 | pages=855–874 | doi-access=free | bibcode=2011JMatS..46..855H }}</ref> They also exhibit different [[melting points]], solubilities, and [[X-ray diffraction]] patterns.

One good example of this is the [[quartz]] form of [[silicon dioxide]], or SiO<sub>2</sub>. In the vast majority of [[silicates]], the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral {SiO<sub>4</sub>} 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 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 {SiO<sub>4</sub>} tetrahedra are shared with others, yielding the net chemical formula for silica: SiO<sub>2</sub>.

Another example is elemental [[tin]] (Sn), which is malleable near ambient temperatures but is [[brittle]] when cooled. This change in mechanical properties due to existence of its two major [[allotrope]]s, α- and β-tin. The two [[allotrope]]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&nbsp;°C and pressures above several GPa.<ref>{{cite journal|first = A. M.|last = Molodets|author2=Nabatov, S. S.|title = Thermodynamic Potentials, Diagram of State, and Phase Transitions of Tin on Shock Compression|journal = High Temperature|volume = 38|issue = 5|year = 2000|pages = 715–721|doi = 10.1007/BF02755923| bibcode=2000HTemp..38..715M |s2cid = 120417927}}</ref> White tin is metallic, and is the stable crystalline form at or above room temperature. Below 13.2&nbsp;°C, tin exists in the gray form, which has a [[diamond cubic]] crystal structure, similar to [[diamond]], [[silicon]] or [[germanium]]. Gray tin has no metallic properties at all, is a dull gray powdery material, and has few uses, other than a few specialized [[semiconductor]] applications.<ref>{{cite book|publisher = Walter de Gruyter|year = 1985|edition = 91–100|pages = 793–800|isbn = 978-3-11-007511-3|title = Lehrbuch der Anorganischen Chemie|first = Arnold F.|last = Holleman |author2=Wiberg, Egon |author3=Wiberg, Nils|chapter = Tin| language = de}}</ref> Although the α–β transformation temperature of tin is nominally 13.2&nbsp;°C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0&nbsp;°C, and upon addition of Sb or Bi the transformation may not occur at all.<ref>{{cite book|first = Mel|last = Schwartz |title = Encyclopedia of Materials, Parts and Finishes|edition = 2nd|chapter = Tin and Alloys, Properties|publisher = CRC Press|year = 2002|isbn= 978-1-56676-661-6}}</ref>

== Physical properties ==
Twenty of the 32 crystal classes are [[piezoelectric]], and crystals belonging to one of these classes (point groups) display [[piezoelectricity]]. All piezoelectric classes lack [[inversion symmetry]]. Any material develops a [[dielectric]] 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 ten of the 32 point groups are [[polar point group|polar]]. All polar crystals are [[pyroelectric]], so the ten polar crystal classes are sometimes referred to as the pyroelectric classes.

There are a few crystal structures, notably the [[Perovskite (structure)|perovskite structure]], which exhibit [[ferroelectric]] behavior. This is analogous to [[ferromagnetism]], 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).

== Some examples of crystal structures ==
{{Gallery
|title=Some examples of crystal structures
|width=160 | height=170
|align=center
|File:Sodium-acetate-trihydrate-chain-from-xtal-3D-bs-17.png 
 |CH<sub>3</sub>COONa.3H<sub>2</sub>O
|File:Sodium-ethoxide-xtal-layer-view-1-Mercury-3D-bs.png 
 |C<sub>2</sub>H<sub>5</sub>ONa
 |alt2=Profile of stone face on mountainside, with 3 workers.
|File:CsCl_polyhedra.png 
 |CsCl
 |alt3=Shiny silver coin with profile of Washington bust. 
|File:Xtals_combined_2_300ppi.png 
 |CaF<sub>2</sub>
 |alt4=Gold coin with bust of Washington facing slightly left
|File:Rutile-unit-cell-3D-balls.png 
 |TiO<sub>2</sub>
 |File:Sodium-chloride-3D-ionic.png 
 |NaCl
 |File:Β-Quartz.svg 
 |β-quartz (SiO<sub>2</sub>)
 |File:Corundum.png 
 |Ruby
 }}

==See also==
{{for|more detailed information in specific technology applications|Materials science|Ceramic engineering|Metallurgy|List of biophysically important macromolecular crystal structures}}
{{div col}}
*[[Brillouin zone]] – a primitive cell in the reciprocal space lattice of a crystal
*[[Crystal engineering]]
*[[Crystal growth]] – a major stage of a crystallization process
*[[Crystallographic database]]
*[[Fractional coordinates]]
*[[Frank–Kasper phases]]
*[[Hermann–Mauguin notation]] – a notation to represent symmetry in point groups, plane groups and space groups
*[[Laser-heated pedestal growth]] – a crystal growth technique
*[[Liquid crystal]] – a state of matter with properties of both conventional liquids and crystals
*[[Patterson function]] – a function used to solve the phase problem in [[X-ray crystallography]]
*[[Periodic table (crystal structure)]] – (for elements that are solid at standard temperature and pressure) gives the crystalline structure of the most thermodynamically stable form(s) in those conditions. In all other cases the structure given is for the element at its melting point.
*[[Primitive cell]] – a repeating unit formed by the vectors spanning the points of a lattice
*[[Seed crystal]] – a small piece of a single crystal used to initiate growth of a larger crystal
*[[Wigner–Seitz cell]] – a primitive cell of a crystal lattice with Voronoi decomposition applied
{{div col end}}

==References==
{{Reflist|30em}}

==External links==
{{Commons-inline}}
*[https://www.xtal.iqf.csic.es/Cristalografia/index-en.html The internal structure of crystals... Crystallography for beginners]
*[https://www.enggstudy.com/basic-types-of-crystal-structure/ Different types of crystal structure]
*[https://web.archive.org/web/20160304130329/http://iffwww.iff.kfa-juelich.de/icp/atoms/atoms.sgml-7.html Appendix A from the manual for Atoms, software for XAFS]
*[https://web.archive.org/web/20060822042901/http://dave.ucsc.edu/myrtreia/crystal.html Intro to Minerals: Crystal Class and System]
*[https://web.archive.org/web/20060826015700/http://www.rockhounds.com/rockshop/xtal/index.html Introduction to Crystallography and Mineral Crystal Systems]
*[https://web.archive.org/web/20060208140059/http://www.ece.byu.edu/cleanroom/EW_orientation.phtml Crystal planes and Miller indices]
*[http://www.ibiblio.org/e-notes/Cryst/Cryst.htm Interactive 3D Crystal models]
*[http://chemannex.weebly.com/crystal-structures.html Specific Crystal 3D models]
*[[Crystallography open database|Crystallography Open Database (with more than 140,000 crystal structures)]]

{{Fundamental aspects of materials science}}
{{Crystallography}}
{{Crystal systems}}
{{Authority control}}

[[Category:Chemical properties]]
[[Category:Crystallography]]
[[Category:Materials science]]
[[Category:Crystals]]
[[Category:Structure]]