Difference between revisions of "Centred lattice"
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== Definition == | == 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<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 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: | ||
+ | |||
+ | ''V<sub>c</sub>'' = ''m V'' | ||
+ | |||
+ | == Types of centred lattices == | ||
+ | {| border="1" cellpadding="1" 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|| 1|| ''V'' | ||
+ | |- | ||
+ | | ''A''|| ''A''-face centred|| ½'''b<sub>c</sub>'''+½'''c<sub>c</sub>'''|| 2|| 2''V'' | ||
+ | |- | ||
+ | | ''B''|| ''B''-face centred|| ½'''c<sub>c</sub>'''+½'''a<sub>c</sub>'''|| 2|| 2''V'' | ||
+ | |- | ||
+ | | ''C''|| ''C''-face centred|| ½'''a<sub>c</sub>'''+½'''b<sub>c</sub>'''|| 2|| 2''V'' | ||
+ | |- | ||
+ | | ''I''|| body centred<br>(''Innenzentriert'')|| ½'''a<sub>c</sub>'''+½'''b<sub>c</sub>'''+½'''c<sub>c</sub>'''|| 2|| 2''V'' | ||
+ | |- | ||
+ | |rowspan=3| ''F'' | ||
+ | |rowspan=3| All-face centred | ||
+ | || ½'''a<sub>c</sub>'''+½'''b<sub>c</sub>''' | ||
+ | |rowspan=3| ''4'' | ||
+ | |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|| 1|| ''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| 3 | ||
+ | |rowspan=2| 3''V'' | ||
+ | |- | ||
+ | || | ||
+ | ⅓'''a<sub>c</sub>'''+⅔'''b<sub>c</sub>'''+⅔'''c<sub>c</sub>''' | ||
+ | |||
+ | |- | ||
+ | |rowspan=2| ''H'' | ||
+ | |rowspan=2| Hexagonally centred | ||
+ | |⅔'''a<sub>c</sub>'''+⅓'''b<sub>c</sub>''' | ||
+ | |rowspan=2| 3 | ||
+ | |rowspan=2| 3''V'' | ||
+ | |- | ||
+ | || ⅓'''a<sub>c</sub>'''+⅔'''b<sub>c</sub>''' | ||
+ | |- | ||
+ | |} | ||
=== See also === | === See also === | ||
− | Sections 1.2 and 9 of ''International Tables of Crystallography, Volume A'' | + | Sections 1.2 and 9 of ''International Tables of Crystallography, Volume A''<br> |
+ | Section 1.1 of ''International Tables of Crystallography, Volume C'' | ||
− | |||
[[Category:Fundamental crystallography]]<br> | [[Category:Fundamental crystallography]]<br> |
Revision as of 14:57, 26 January 2006
Contents
Centred lattices
Other languages
Réseaux centrés (Fr). redes centradas (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
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 | 1 | V |
A | A-face centred | ½b_{c}+½c_{c} | 2 | 2V |
B | B-face centred | ½c_{c}+½a_{c} | 2 | 2V |
C | C-face centred | ½a_{c}+½b_{c} | 2 | 2V |
I | body centred (Innenzentriert) |
½a_{c}+½b_{c}+½c_{c} | 2 | 2V |
F | All-face centred | ½a_{c}+½b_{c} | 4 | 4V |
½b_{c}+½c_{c} | ||||
½c_{c}+½a_{c} | ||||
R | Primitive (rhombohedral axes) |
0 | 1 | V |
R | Rhombohedrally centred (hexagonal axes) |
⅔a_{c}+⅓b_{c}+⅓c_{c} | 3 | 3V |
⅓a_{c}+⅔b_{c}+⅔c_{c} | ||||
H | Hexagonally centred | ⅔a_{c}+⅓b_{c} | 3 | 3V |
⅓a_{c}+⅔b_{c} |
See also
Sections 1.2 and 9 of International Tables of Crystallography, Volume A
Section 1.1 of International Tables of Crystallography, Volume C