Difference between revisions of "Centred lattice"
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− | = | + | <font color="blue">Réseau centré</Font> (''Fr''). <font color="red">Zentriertes Gitter</font> (''Ge''). <font color="black">Reticoli centrati</font> (''It''). <font color="purple">複合格子</Font> (''Ja''). <font color="green">Red centrada</font> (''Sp''). |
− | == | + | == Definition == |
− | + | When the unit cell does not reflect the symmetry of the lattice, it is usual in crystallography to refer to a '[[conventional cell|conventional]]', non-primitive, crystallographic basis, '''a<sub>c</sub>''', '''b<sub>c</sub>''', '''c<sub>c</sub>''' instead of a [[primitive_cell| primitive basis]], '''a''', '''b''', '''c'''. This is done by adding lattice nodes at the center of the unit cell or at one or three faces. The vectors joining the origin of the unit cell to these additional nodes are called 'centring vectors'. In such a lattice, '''a<sub>c</sub>''', '''b<sub>c</sub>''' and '''c<sub>c</sub>''' with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors '''t''' ∈ '''L''', '''t''' = ''t<sub>1</sub>'' '''a<sub>c</sub>''' + ''t<sub>2</sub>'' '''b<sub>c</sub>''' + ''t<sub>3</sub>'' '''c<sub>c</sub>'''; with at least two of the coefficients ''t<sub>1</sub>'', ''t<sub>2</sub>'', ''t<sub>3</sub>'' being fractional. The table below gives the various types of centring vectors and the corresponding types of centring. Each one is described by a letter, called the Bravais letter, which is to be found in the [[Hermann-Mauguin symbols|Hermann-Mauguin symbol]] of a space group. | |
+ | The 'multiplicity', ''m'', of the centred cell is the number of lattice nodes per unit cell (see table). | ||
+ | |||
+ | The [[unit_cell|volume of the unit cell]], ''V<sub>c</sub>'' = ('''a<sub>c</sub>''', '''b<sub>c</sub>''', '''c<sub>c</sub>''') is given in terms of the volume of the primitive cell, ''V'' = ('''a''', '''b''', '''c'''), by | ||
+ | <div align="center"> | ||
+ | ''V<sub>c</sub>'' = ''m V.'' | ||
+ | </div> | ||
+ | ==Centred cells vs 'centred lattices'== | ||
+ | A lattice being an infinite, symmetric and periodic collection of zero-dimensional nodes, rigorously speaking it is neither primitive nor centred. The expression 'centred lattice' has to be considered as a shortcut for 'lattice whose [[conventional cell]] is centred'. | ||
+ | |||
+ | == Types of centred lattices == | ||
+ | {| border="1" cellpadding="2" cellspacing="2" align="center" | ||
+ | ! Bravais letter !! Centring type !! Centring vectors !! Multiplicity<br> (number of nodes per unit cell)!! Unit-cell volume <math>V_c</math> | ||
+ | |- | ||
+ | | ''P''|| Primitive|| 0|| <center>1</center>|| ''V'' | ||
+ | |- | ||
+ | | ''A''|| ''A''-face centred|| ½'''b<sub>c</sub>'''+½'''c<sub>c</sub>'''|| <center>2</center>|| 2''V'' | ||
+ | |- | ||
+ | | ''B''|| ''B''-face centred|| ½'''c<sub>c</sub>'''+½'''a<sub>c</sub>'''|| <center>2</center>|| 2''V'' | ||
+ | |- | ||
+ | | ''C''|| ''C''-face centred|| ½'''a<sub>c</sub>'''+½'''b<sub>c</sub>'''|| <center>2</center>|| 2''V'' | ||
+ | |- | ||
+ | | ''I''|| body centred<br>(''Innenzentriert'')|| ½'''a<sub>c</sub>'''+½'''b<sub>c</sub>'''+½'''c<sub>c</sub>'''|| <center>2</center>|| 2''V'' | ||
+ | |- | ||
+ | |rowspan=3| ''F'' | ||
+ | |rowspan=3| All-face centred | ||
+ | || ½'''a<sub>c</sub>'''+½'''b<sub>c</sub>''' | ||
+ | |rowspan=3|<center>4</center> | ||
+ | |rowspan=3| 4''V'' | ||
+ | |- | ||
+ | || ½'''b<sub>c</sub>'''+½'''c<sub>c</sub>''' | ||
+ | |- | ||
+ | || ½'''c<sub>c</sub>'''+½'''a<sub>c</sub>''' | ||
+ | |- | ||
+ | | ''R''|| Primitive<br>(rhombohedral axes)|| 0|| <center>1</center>|| ''V'' | ||
+ | |- | ||
+ | |rowspan=2| ''R'' | ||
+ | |rowspan=2| Rhombohedrally centred<br>(hexagonal axes) | ||
+ | | ⅔'''a<sub>c</sub>'''+⅓'''b<sub>c</sub>'''+⅓'''c<sub>c</sub>''' | ||
+ | |rowspan=2| <center>3</center> | ||
+ | |rowspan=2| 3''V'' | ||
+ | |- | ||
+ | || | ||
+ | ⅓'''a<sub>c</sub>'''+⅔'''b<sub>c</sub>'''+⅔'''c<sub>c</sub>''' | ||
+ | |||
+ | |- | ||
+ | |rowspan=2| ''[[H centred cell|H]]'' | ||
+ | |rowspan=2| [[H centred cell|Hexagonally centred]] | ||
+ | |⅔'''a<sub>c</sub>'''+⅓'''b<sub>c</sub>''' | ||
+ | |rowspan=2| <center>3</center> | ||
+ | |rowspan=2| 3''V'' | ||
+ | |- | ||
+ | || ⅓'''a<sub>c</sub>'''+⅔'''b<sub>c</sub>''' | ||
+ | |- | ||
− | |||
− | < | + | |- |
+ | |rowspan=2| ''[[D centred cell|D]]'' | ||
+ | |rowspan=2| [[D centred cell|Rhombohedrally centred]] | ||
+ | |⅓'''a<sub>c</sub>'''+⅓'''b<sub>c</sub>'''+⅓'''c<sub>c</sub>''' | ||
+ | |rowspan=2| <center>3</center> | ||
+ | |rowspan=2| 3''V'' | ||
+ | |- | ||
+ | || ⅔'''a<sub>c</sub>'''+⅔'''b<sub>c</sub>'''+⅔'''c<sub>c</sub>''' | ||
+ | |- | ||
+ | |} | ||
− | + | The letter ''S'' is also used to indicate a single pair of centred faces. This happens in the monoclinic and orthorhombic [[crystal family|crystal families]]. | |
+ | *In the monoclinic crystal family, ''b''-unique axis, the centred cells ''mA'', ''mC'', ''mI'' and ''mF'' are equivalent in the sense that a different choice of axes in the (010) plane interchanges these centrings. The symbol ''mS'' is sometimes used to collectively indicate any of these cells. The cell ''mB'' is instead equivalent to the cell ''mP''. | ||
+ | *In the orthorhombic crystal family, the centred cells ''oA'', ''oB'' and ''oC'' are transformed one into the other when the axes are permuted. The symbol ''oS'' is sometimes used to collectively indicate these three equivalent cells. | ||
− | + | == See also == | |
+ | *[[Orthohexagonal]] | ||
+ | *Chapter 1.3.2.4 of ''International Tables for Crystallography, Volume A'', 6th edition | ||
+ | *Chapter 1.1 of ''International Tables for Crystallography, Volume C'' | ||
+ | *[http://pubs.acs.org/doi/abs/10.1021/ed070p959 Discussion about centred lattices ''vs'' centred cells] | ||
− | |||
[[Category:Fundamental crystallography]]<br> | [[Category:Fundamental crystallography]]<br> |
Latest revision as of 16:46, 2 February 2018
Réseau centré (Fr). Zentriertes Gitter (Ge). Reticoli centrati (It). 複合格子 (Ja). Red centrada (Sp).
Definition
When the unit cell does not reflect the symmetry of the lattice, it is usual in crystallography to refer to a 'conventional', non-primitive, crystallographic basis, a_{c}, b_{c}, c_{c} instead of a primitive basis, a, b, c. This is done by adding lattice nodes at the center of the unit cell or at one or three faces. The vectors joining the origin of the unit cell to these additional nodes are called 'centring vectors'. In such a lattice, a_{c}, b_{c} and c_{c} with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors t ∈ L, t = t_{1} a_{c} + t_{2} b_{c} + t_{3} c_{c}; with at least two of the coefficients t_{1}, t_{2}, t_{3} being fractional. The table below gives the various types of centring vectors and the corresponding types of centring. Each one is described by a letter, called the Bravais letter, which is to be found in the Hermann-Mauguin symbol of a space group.
The 'multiplicity', m, of the centred cell is the number of lattice nodes per unit cell (see table).
The volume of the unit cell, V_{c} = (a_{c}, b_{c}, c_{c}) is given in terms of the volume of the primitive cell, V = (a, b, c), by
V_{c} = m V.
Centred cells vs 'centred lattices'
A lattice being an infinite, symmetric and periodic collection of zero-dimensional nodes, rigorously speaking it is neither primitive nor centred. The expression 'centred lattice' has to be considered as a shortcut for 'lattice whose conventional cell is centred'.
Types of centred lattices
Bravais letter | Centring type | Centring vectors | Multiplicity (number of nodes per unit cell) |
Unit-cell volume [math]V_c[/math] |
---|---|---|---|---|
P | Primitive | 0 | |
V |
A | A-face centred | ½b_{c}+½c_{c} | |
2V |
B | B-face centred | ½c_{c}+½a_{c} | |
2V |
C | C-face centred | ½a_{c}+½b_{c} | |
2V |
I | body centred (Innenzentriert) |
½a_{c}+½b_{c}+½c_{c} | |
2V |
F | All-face centred | ½a_{c}+½b_{c} | 4V | |
½b_{c}+½c_{c} | ||||
½c_{c}+½a_{c} | ||||
R | Primitive (rhombohedral axes) |
0 | |
V |
R | Rhombohedrally centred (hexagonal axes) |
⅔a_{c}+⅓b_{c}+⅓c_{c} | |
3V |
⅓a_{c}+⅔b_{c}+⅔c_{c} | ||||
H | Hexagonally centred | ⅔a_{c}+⅓b_{c} | |
3V |
⅓a_{c}+⅔b_{c} | ||||
D | Rhombohedrally centred | ⅓a_{c}+⅓b_{c}+⅓c_{c} | |
3V |
⅔a_{c}+⅔b_{c}+⅔c_{c} |
The letter S is also used to indicate a single pair of centred faces. This happens in the monoclinic and orthorhombic crystal families.
- In the monoclinic crystal family, b-unique axis, the centred cells mA, mC, mI and mF are equivalent in the sense that a different choice of axes in the (010) plane interchanges these centrings. The symbol mS is sometimes used to collectively indicate any of these cells. The cell mB is instead equivalent to the cell mP.
- In the orthorhombic crystal family, the centred cells oA, oB and oC are transformed one into the other when the axes are permuted. The symbol oS is sometimes used to collectively indicate these three equivalent cells.
See also
- Orthohexagonal
- Chapter 1.3.2.4 of International Tables for Crystallography, Volume A, 6th edition
- Chapter 1.1 of International Tables for Crystallography, Volume C
- Discussion about centred lattices vs centred cells