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In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice.^[1] Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessarily have unit size, or even a particular size at all. Rather, the primitive cell is the closest analogy to a unit vector, since it has a determined size for a given lattice and is the basic building block from which larger cells are constructed.

The concept is used particularly in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its unit cell, which is a section of the tiling (a parallelogram or parallelepiped) that generates the whole tiling using only translations.

There are two special cases of the unit cell: the primitive cell and the conventional cell. The primitive cell is a unit cell corresponding to a single lattice point, it is the smallest possible unit cell.^[2] In some cases, the full symmetry of a crystal structure is not obvious from the primitive cell, in which cases a conventional cell may be used. A conventional cell (which may or may not be primitive) is a unit cell with the full symmetry of the lattice and may include more than one lattice point. The conventional unit cells are parallelotopes in n dimensions.

Primitive cell

A primitive cell is a unit cell that contains exactly one lattice point. For unit cells generally, lattice points that are shared by $n$ cells are counted as ⁠1/ $n$ ⁠ of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain ⁠1/8⁠ of each of them.^[3] An alternative conceptualization is to consistently pick only one of the $n$ lattice points to belong to the given unit cell (so the other $n-1$ lattice points belong to adjacent unit cells).

The primitive translation vectors $a \to 1$ , $a \to 2$ , $a \to 3$ span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector

{\vec {T}}=u_{1}{\vec {a}}_{1}+u_{2}{\vec {a}}_{2}+u_{3}{\vec {a}}_{3},

where $u 1$ , $u 2$ , $u 3$ are integers, translation by which leaves the lattice invariant.^{[note 1]} That is, for a point in the lattice $r$ , the arrangement of points appears the same from $r' = r + T \to$ as from $r$ .^[4]

Since the primitive cell is defined by the primitive axes (vectors) $a \to 1$ , $a \to 2$ , $a \to 3$ , the volume $V p$ of the primitive cell is given by the parallelepiped from the above axes as

V_{\mathrm {p} }=\left|{\vec {a}}_{1}\cdot ({\vec {a}}_{2}\times {\vec {a}}_{3})\right|.

Usually, primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds, with an atom at each corner of the cell. This choice of primitive cell is not unique, but volume of primitive cells will always be given by the expression above.^[5]

Wigner–Seitz cell

In addition to the parallelepiped primitive cells, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of Voronoi cell. The Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone.

Conventional cell

For each particular lattice, a conventional cell has been chosen on a case-by-case basis by crystallographers based on convenience of calculation.^[6] These conventional cells may have additional lattice points located in the middle of the faces or body of the unit cell. The number of lattice points, as well as the volume of the conventional cell is an integer multiple (1, 2, 3, or 4) of that of the primitive cell.^[7]

Two dimensions

For any 2-dimensional lattice, the unit cells are parallelograms, which in special cases may have orthogonal angles, equal lengths, or both. Four of the five two-dimensional Bravais lattices are represented using conventional primitive cells, as shown below.

Conventional primitive cell
Shape name	Parallelogram	Rectangle	Square	Rhombus
Bravais lattice	Primitive Oblique	Primitive Rectangular	Primitive Square	Primitive Hexagonal

The centered rectangular lattice also has a primitive cell in the shape of a rhombus, but in order to allow easy discrimination on the basis of symmetry, it is represented by a conventional cell which contains two lattice points.

Primitive cell
Shape name	Rhombus
Conventional cell
Bravais lattice	Centered Rectangular

Three dimensions

For any 3-dimensional lattice, the conventional unit cells are parallelepipeds, which in special cases may have orthogonal angles, or equal lengths, or both. Seven of the fourteen three-dimensional Bravais lattices are represented using conventional primitive cells, as shown below.

Conventional primitive cell
Shape name	Parallelepiped	Oblique rectangular prism	Rectangular cuboid	Square cuboid	Trigonal trapezohedron	Cube	Right rhombic prism
Bravais lattice	Primitive Triclinic	Primitive Monoclinic	Primitive Orthorhombic	Primitive Tetragonal	Primitive Rhombohedral	Primitive Cubic	Primitive Hexagonal

The other seven Bravais lattices (known as the centered lattices) also have primitive cells in the shape of a parallelepiped, but in order to allow easy discrimination on the basis of symmetry, they are represented by conventional cells which contain more than one lattice point.

Primitive cell
Shape name	Oblique rhombic prism	Right rhombic prism
Conventional cell
Bravais lattice	Base-centered Monoclinic	Base-centered Orthorhombic	Body-centered Orthorhombic	Face-centered Orthorhombic	Body-centered Tetragonal	Body-centered Cubic	Face-centered Cubic

Notes

^ In $n$ dimensions the crystal translation vector would be
${\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.$
That is, for a point in the lattice $r$ , the arrangement of points appears the same from $r' = r + T \to$ as from $r$ .

References

^ Ashcroft, Neil W. (1976). "Chapter 4". Solid State Physics. W. B. Saunders Company. p. 72. ISBN 0-03-083993-9.
^ Simon, Steven (2013). The Oxford Solid State Physics (1 ed.). Oxford University Press. p. 114. ISBN 978-0-19-968076-4.
^ "DoITPoMS – TLP Library Crystallography – Unit Cell". Online Materials Science Learning Resources: DoITPoMS. University of Cambridge. Retrieved 21 February 2015.
^ Kittel, Charles (11 November 2004). Introduction to Solid State Physics (8 ed.). Wiley. p. 4. ISBN 978-0-471-41526-8.
^ Mehl, Michael J.; Hicks, David; Toher, Cormac; Levy, Ohad; Hanson, Robert M.; Hart, Gus; Curtarolo, Stefano (2017). "The AFLOW Library of Crystallographic Prototypes: Part 1". Computational Materials Science. 136. Elsevier BV: S1–S828. arXiv:1806.07864. doi:10.1016/j.commatsci.2017.01.017. ISSN 0927-0256. S2CID 119490841.
^ Aroyo, M. I., ed. (2016-12-31). International Tables for Crystallography. Chester, England: International Union of Crystallography. p. 25. doi:10.1107/97809553602060000114. ISBN 978-0-470-97423-0.
^ Ashcroft, Neil W. (1976). Solid State Physics. W. B. Saunders Company. p. 73. ISBN 0-03-083993-9.

[first-4] In $n$ dimensions the crystal translation vector would be
${\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.$
That is, for a point in the lattice $r$ , the arrangement of points appears the same from $r' = r + T \to$ as from $r$ .

[1] Ashcroft, Neil W. (1976). "Chapter 4". Solid State Physics. W. B. Saunders Company. p. 72. ISBN 0-03-083993-9.

[2] Simon, Steven (2013). The Oxford Solid State Physics (1 ed.). Oxford University Press. p. 114. ISBN 978-0-19-968076-4.

[doitpoms-3] "DoITPoMS – TLP Library Crystallography – Unit Cell". Online Materials Science Learning Resources: DoITPoMS. University of Cambridge. Retrieved 21 February 2015.

[5] Kittel, Charles (11 November 2004). Introduction to Solid State Physics (8 ed.). Wiley. p. 4. ISBN 978-0-471-41526-8.

[6] Mehl, Michael J.; Hicks, David; Toher, Cormac; Levy, Ohad; Hanson, Robert M.; Hart, Gus; Curtarolo, Stefano (2017). "The AFLOW Library of Crystallographic Prototypes: Part 1". Computational Materials Science. 136. Elsevier BV: S1–S828. arXiv:1806.07864. doi:10.1016/j.commatsci.2017.01.017. ISSN 0927-0256. S2CID 119490841.

[7] Aroyo, M. I., ed. (2016-12-31). International Tables for Crystallography. Chester, England: International Union of Crystallography. p. 25. doi:10.1107/97809553602060000114. ISBN 978-0-470-97423-0.

[8] Ashcroft, Neil W. (1976). Solid State Physics. W. B. Saunders Company. p. 73. ISBN 0-03-083993-9.

[1]

[2]

[3]

[note 1]

[4]

[5]

[6]

[7]

@@ Line 1: / Line 1: @@
+{{refimprove|date=May 2021}}
-{{short description|Repeating unit formed by the vectors spanning the points of a lattice}}
+{{short description|Repeating unit formed by the vectors spanning the points of a lattice}}
-In [[geometry]], [[biology]], [[mineralogy]], and [[solid state physics]],  a '''unit cell''' is a repeating unit formed by the vectors spanning the points of a lattice.<ref>{{cite book |last=Ashcroft |first=Neil W. |title=Solid State Physics |year=1976 |publisher=W. B. Saunders Company|isbn=0-03-083993-9 |pages=72 |chapter=Chapter 4}}</ref> The geometry of the unit cell is defined as a [[parallelotope]] in n dimensions. That is, the set of all points '''R''' of the form:
+In [[geometry]], [[biology]], [[mineralogy]] and [[solid state physics]],  a '''unit cell''' is a repeating unit formed by the vectors spanning the points of a lattice.<ref>{{cite book |last=Ashcroft |first=Neil W. |title=Solid State Physics |year=1976 |publisher=W. B. Saunders Company|isbn=0-03-083993-9 |pages=72 |chapter=Chapter 4}}</ref> Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessarily have unit size, or even a particular size at all. Rather, the primitive cell is the closest analogy to a unit vector, since it has a determined size for a given lattice and is the basic building block from which larger cells are constructed.
-:<math>\mathbf{R} = x_{1}\mathbf{a}_{1} + x_{2}\mathbf{a}_{2} + \ldots</math>
-The concept is used particularly in describing [[crystal structure]] in two and three dimensions, though it makes sense in all dimensions.  A lattice can be characterized by the geometry of its unit cell. The unit cell is a section of the tiling (a [[parallelogram]] or [[parallelepiped]]) that generates the whole tiling using only translations, and is as small as possible.
+The concept is used particularly in describing [[crystal structure]] in two and three dimensions, though it makes sense in all dimensions.  A lattice can be characterized by the geometry of its unit cell, which is a section of the tiling (a [[parallelogram]] or [[parallelepiped]]) that generates the whole tiling using only translations.
-There are two special cases of the unit cell: the '''primitive cell''' and the '''conventional cell'''. The primitive cell is a unit cell corresponding to a single [[lattice point]]. In some cases, the full symmetry of a crystal structure is not obvious from the primitive cell, in which cases a conventional cell may be used. A conventional cell (which may or may not be primitive) is the smallest unit cell with the full symmetry of the lattice and may include more than one lattice point.
+There are two special cases of the unit cell: the '''primitive cell''' and the '''conventional cell'''. The primitive cell is a unit cell corresponding to a single [[lattice point]], it is the smallest possible unit cell.<ref>{{cite book |last1=Simon |first1=Steven |title=The Oxford Solid State Physics |date=2013 |publisher=Oxford University Press |isbn=978-0-19-968076-4 |page=114 |edition=1}}</ref> In some cases, the full symmetry of a crystal structure is not obvious from the primitive cell, in which cases a conventional cell may be used. A conventional cell (which may or may not be primitive) is a unit cell with the full symmetry of the lattice and may include more than one lattice point. The conventional unit cells are [[Parallelepiped#Parallelotope|parallelotopes]] in ''n'' dimensions.
 ==Primitive cell==
-A primitive cell is a unit cell that contains exactly one and only one lattice point.  For unit cells generally, lattice points that are shared by {{mvar|n}} cells are counted as {{sfrac|1|{{mvar|n}}}} of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain {{sfrac|1|8}} of each of them.<ref name=doitpoms>{{cite web|title=DoITPoMS – TLP Library Crystallography – Unit Cell|url=http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php|website=Online Materials Science Learning Resources: DoITPoMS|publisher=University of Cambridge|access-date=21 February 2015|ref=doitpoms}}</ref> An alternative conceptualization is to consistently pick only one of the {{mvar|n}} lattice points to belong to the given unit cell (so the other {{mvar|1-n}} lattice points belong to adjacent unit cells).
+A primitive cell is a unit cell that contains exactly one lattice point.  For unit cells generally, lattice points that are shared by {{mvar|n}} cells are counted as {{sfrac|1|{{mvar|n}}}} of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain {{sfrac|1|8}} of each of them.<ref name=doitpoms>{{cite web|title=DoITPoMS – TLP Library Crystallography – Unit Cell|url=http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php|website=Online Materials Science Learning Resources: DoITPoMS|publisher=University of Cambridge|access-date=21 February 2015|ref=doitpoms}}</ref> An alternative conceptualization is to consistently pick only one of the {{mvar|n}} lattice points to belong to the given unit cell (so the other {{mvar|n-1}} lattice points belong to adjacent unit cells).
 The ''primitive translation vectors'' {{math|{{vec|''a''}}<sub>1</sub>}}, {{math|{{vec|''a''}}<sub>2</sub>}}, {{math|{{vec|''a''}}<sub>3</sub>}} span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector
@@ Line 20: / Line 20: @@
 In {{mvar|n}} dimensions the crystal translation vector would be
 :<math> \vec T = \sum_{i=1}^{n} u_i\vec a_i, \quad \mbox{where }u_i \in \mathbb{Z} \quad \forall i.</math>
-That is, for a point in the lattice {{math|'''r'''}}, the arrangement of points appears the same from {{math|'''r′''' {{=}} '''r''' + {{vec|''T''}}}} as from {{math|'''r'''}}.}} That is, for a point in the lattice {{math|'''r'''}}, the arrangement of points appears the same from {{math|'''r′''' {{=}} '''r''' + {{vec|''T''}}}} as from {{math|'''r'''}}.<ref>{{cite book |last=Kittel |first=Charles |title=[[Introduction to Solid State Physics]] |publisher=Wiley |page=[https://archive.org/details/isbn_9780471415268/page/4 4] |isbn=978-0-471-41526-8 |edition=8 }}</ref>
+That is, for a point in the lattice {{math|'''r'''}}, the arrangement of points appears the same from {{math|'''r′''' {{=}} '''r''' + {{vec|''T''}}}} as from {{math|'''r'''}}.}} That is, for a point in the lattice {{math|'''r'''}}, the arrangement of points appears the same from {{math|'''r′''' {{=}} '''r''' + {{vec|''T''}}}} as from {{math|'''r'''}}.<ref>{{cite book |last=Kittel |first=Charles |title=[[Introduction to Solid State Physics]] |date=11 November 2004 |publisher=Wiley |page=[https://archive.org/details/isbn_9780471415268/page/4 4] |isbn=978-0-471-41526-8 |edition=8 }}</ref>
 Since the primitive cell is defined by the primitive axes (vectors) {{math|{{vec|''a''}}<sub>1</sub>}}, {{math|{{vec|''a''}}<sub>2</sub>}}, {{math|{{vec|''a''}}<sub>3</sub>}}, the volume {{math|''V''<sub>p</sub>}} of the primitive cell is given by the [[parallelepiped]] from the above axes as
-:<math> V_\mathrm{p} = \left| \vec a_1 \cdot ( \vec a_2 \times \vec a_3 ) \right|.</math>
+:         <math> V_\mathrm{p} = \left| \vec a_1 \cdot ( \vec a_2 \times \vec a_3 ) \right|.</math>
+Usually, primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds, with an atom at each corner of the cell. This choice of primitive cell is not unique, but volume of primitive cells will always be given by the expression above.<ref>{{cite journal | last1=Mehl | first1=Michael J. | last2=Hicks | first2=David | last3=Toher | first3=Cormac | last4=Levy | first4=Ohad | last5=Hanson | first5=Robert M. | last6=Hart | first6=Gus | last7=Curtarolo | first7=Stefano | title=The AFLOW Library of Crystallographic Prototypes: Part 1 | journal=Computational Materials Science | publisher=Elsevier BV | volume=136 | year=2017 | issn=0927-0256 | doi=10.1016/j.commatsci.2017.01.017 | pages=S1–S828| arxiv=1806.07864 | s2cid=119490841 }}</ref>
+===Wigner–Seitz cell===
+{{main|Wigner–Seitz cell}}
+In addition to the parallelepiped primitive cells, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of [[Voronoi cell]]. The Wigner–Seitz cell of the [[reciprocal lattice]] in [[momentum space]] is called the [[Brillouin zone]].
 == Conventional cell ==
-A conventional cell is the smallest unit cell whose axes follow the symmetry axes of the crystal structure. Such conventional cells may have additional lattice points located in the middle of the faces or body of the unit cell. The number of lattice points, as well as the volume, of the conventional cell is an integer multiple (typically 1, 2, 3, or 4) of that of the primitive cell.<ref>{{cite book |last=Ashcroft |first=Neil W. |date=1976 |title=Solid State Physics |url=https://archive.org/details/solidstatephysic00ashc?q=just+fills+all+of+space+without+either+overlapping+itself+or+leaving+voids+is+called+a+primitive+cell |publisher=W. B. Saunders Company |page= 73|isbn=0-03-083993-9 |url-access=registration }}</ref>
+For each particular lattice, a conventional cell has been chosen on a case-by-case basis by crystallographers based on convenience of calculation.<ref>{{cite book | editor-last=Aroyo | editor-first=M. I. | title=International Tables for Crystallography | publisher=International Union of Crystallography | publication-place=Chester, England | date=2016-12-31 | isbn=978-0-470-97423-0 | doi=10.1107/97809553602060000114 | page=25}}</ref> These conventional cells may have additional lattice points located in the middle of the faces or body of the unit cell. The number of lattice points, as well as the volume of the conventional cell is an integer multiple (1, 2, 3, or 4) of that of the primitive cell.<ref>{{cite book |last=Ashcroft |first=Neil W. |date=1976 |title=Solid State Physics |url=https://archive.org/details/solidstatephysic00ashc?q=just+fills+all+of+space+without+either+overlapping+itself+or+leaving+voids+is+called+a+primitive+cell |publisher=W. B. Saunders Company |page= 73|isbn=0-03-083993-9 |url-access=registration }}</ref>
 ==Two dimensions==
 [[File:Fundamental parallelogram.png|thumb|The [[parallelogram]] is the general primitive cell for the plane.]]
-For any 2-dimensional lattice, you can find unit cells which are [[parallelogram]]s, which in special cases may have orthogonal angles, or equal lengths, or both.  Some of the five two-dimensional [[Bravais lattice]]s are represented using conventional primitive cells, as shown below.
+For any 2-dimensional lattice, the unit cells are [[parallelogram]]s, which in special cases may have orthogonal angles, equal lengths, or both.  Four of the five two-dimensional [[Bravais lattice]]s are represented using conventional primitive cells, as shown below.
 {| class=wikitable
@@ Line 38: / Line 44: @@
 |[[File:2d op rectangular.svg|100px]]
 |[[File:2d tp.svg|80px]]
+|[[File:2d hp.svg|100px]]
 |-
 ! Shape name
@@ Line 43: / Line 50: @@
 |[[Rectangle]]
 |[[Square]]
+|[[Rhombus]]
 |-
 ! Bravais lattice
@@ Line 48: / Line 56: @@
 |Primitive Rectangular
 |Primitive Square
+|Primitive Hexagonal
 |}
@@ Line 68: / Line 77: @@
 ==Three dimensions==
 [[File:Parallelepiped 2013-11-29.svg|thumb|A [[parallelepiped]] is a general primitive cell for 3-dimensional space.]]
-For any 3-dimensional lattice, you can find unit cells which are [[parallelepiped]]s, which in special cases may have orthogonal angles, or equal lengths, or both. Some of the fourteen three-dimensional [[Bravais lattice]]s are represented using conventional primitive cells, as shown below.
+For any 3-dimensional lattice, the conventional unit cells are [[parallelepiped]]s, which in special cases may have orthogonal angles, or equal lengths, or both. Seven of the fourteen three-dimensional [[Bravais lattice]]s are represented using conventional primitive cells, as shown below.
 {| class=wikitable
@@ Line 78: / Line 87: @@
 |[[File:Rhombohedral.svg|100px]]
 |[[File:Cubic.svg|100px]]
+|[[File:Hexagonal latticeFRONT.svg|80px|Hexagonal]]
 |-
 ! Shape name
@@ Line 86: / Line 96: @@
 |[[Trigonal trapezohedron]]
 |[[Cube]]
+|Right rhombic [[Prism (geometry)|prism]]
 |-
 ! Bravais lattice
@@ Line 94: / Line 105: @@
 |Primitive [[Rhombohedral lattice system|Rhombohedral]]
 |Primitive [[Cubic crystal system|Cubic]]
+|Primitive [[Hexagonal crystal system|Hexagonal]]
 |}
-The other Bravais lattices also have primitive cells in the shape of a parallelepiped, but in order to allow easy discrimination on the basis of symmetry, they are represented by conventional cells which contain more than one lattice point.
+The other seven Bravais lattices (known as the centered lattices) also have primitive cells in the shape of a parallelepiped, but in order to allow easy discrimination on the basis of symmetry, they are represented by conventional cells which contain more than one lattice point.
 {| class=wikitable
@@ Line 102: / Line 114: @@
 |[[File:Clinorhombic prism.svg|100px]]
 |[[File:Rhombic prism.svg|80px]]
+|
+|
+|
+|
+|
 |-
 ! Shape name
 |Oblique rhombic [[Prism (geometry)|prism]]
 |Right rhombic [[Prism (geometry)|prism]]
+|
+|
+|
+|
+|
 |-
 ! Conventional cell
-|[[File:Monoclinic-base-centered.svg|80px]]
+|[[File:Base-centered monoclinic.svg|80px]]
 |[[File:Orthorhombic-base-centered.svg|80px]]
+|[[File:Orthorhombic-body-centered.svg|80px]]
+|[[File:Orthorhombic-face-centered.svg|80px]]
+|[[File:Tetragonal-body-centered.svg|80px]]
+|[[File:Cubic-body-centered.svg|80px]]
+|[[File:Cubic-face-centered.svg|80px]]
 |-
 ! Bravais lattice
 |Base-centered [[Monoclinic crystal system|Monoclinic]]
 |Base-centered [[Orthorhombic crystal system|Orthorhombic]]
+|Body-centered [[Orthorhombic crystal system|Orthorhombic]]
+|Face-centered [[Orthorhombic crystal system|Orthorhombic]]
+|Body-centered [[Tetragonal crystal system|Tetragonal]]
+|Body-centered [[Cubic crystal system|Cubic]]
+|Face-centered [[Cubic crystal system|Cubic]]
 |}
-==Wigner–Seitz cell==
-{{main|Wigner–Seitz cell}}
-An alternative to the unit cell, for every Bravais lattice there is another kind of cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of [[Voronoi cell]]. The Wigner–Seitz cell of the [[reciprocal lattice]] in [[momentum space]] is called the [[Brillouin zone]].
 == See also ==
@@ Line 131: / Line 159: @@
 ==References==
 {{Reflist}}
+{{Crystallography}}
 [[Category:Crystallography]]