Centred lattice

(Difference between revisions)
 Revision as of 12:38, 25 January 2006 (view source)← Older edit Revision as of 14:57, 26 January 2006 (view source)Newer edit → Line 8: Line 8: == Definition == == Definition == - Provide the definition of the entry (in English) here. + 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, '''ac''', '''bc''', '''cc''' 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 '''ac''', '''bc''' and '''cc''' with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors '''t''' ∈ '''L''', '''t''' = ''t1'' '''ac''' + ''t2'' '''bc''' + ''t3'' '''cc'''; with at least two of the coefficients '''t1''', '''t2''', '''t3''' 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]], ''Vc'' = ('''ac''', '''bc''', '''cc''') is given in terms of the volume of the primitive cell, ''V'' = ('''a''', '''b''', '''c'''), by: + + ''Vc'' = ''m V'' + + == Types of centred lattices == + {| border="1" cellpadding="1" align="center" + | Bravais letter || Centring type || Centring vectors || Multiplicity
(number of nodes per unit cell)|| Unit-cell volume [itex]V_c[/itex] + |- + | ''P''|| Primitive|| 0|| 1|| ''V'' + |- + |  ''A''|| ''A''-face centred|| ½'''bc'''+½'''cc'''|| 2|| 2''V'' + |- + |  ''B''|| ''B''-face centred|| ½'''cc'''+½'''ac'''|| 2|| 2''V'' + |- + |  ''C''|| ''C''-face centred|| ½'''ac'''+½'''bc'''|| 2|| 2''V'' + |- + |  ''I''|| body centred
(''Innenzentriert'')|| ½'''ac'''+½'''bc'''+½'''cc'''|| 2|| 2''V'' + |- + |rowspan=3| ''F'' + |rowspan=3| All-face centred + || ½'''ac'''+½'''bc''' + |rowspan=3| ''4'' + |rowspan=3| 4''V'' + |- + ||  ½'''bc'''+½'''cc''' + |- + ||  ½'''cc'''+½'''ac''' + |- + |  ''R''|| Primitive
(rhombohedral axes)|| 0|| 1|| ''V'' + |- + |rowspan=2|  ''R'' + |rowspan=2| Rhombohedrally centred
(hexagonal axes) + | ⅔'''ac'''+⅓'''bc'''+⅓'''cc''' + |rowspan=2| 3 + |rowspan=2| 3''V'' + |- + || + ⅓'''ac'''+⅔'''bc'''+⅔'''cc''' + + |- + |rowspan=2|  ''H'' + |rowspan=2| Hexagonally centred + |⅔'''ac'''+⅓'''bc''' + |rowspan=2| 3 + |rowspan=2| 3''V'' + |- + ||  ⅓'''ac'''+⅔'''bc''' + |- + |} === 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''
+ Section 1.1 of ''International Tables of Crystallography, Volume C'' - ---- [[Category:Fundamental crystallography]]
[[Category:Fundamental crystallography]]

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, ac, bc, cc 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 ac, bc and cc with all their integral linear combinations are lattice vectors again, but there exist other lattice vectors tL, t = t1 ac + t2 bc + t3 cc; with at least two of the coefficients t1, t2, t3 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, Vc = (ac, bc, cc) is given in terms of the volume of the primitive cell, V = (a, b, c), by:

Vc = m V

Types of centred lattices

 Bravais letter Centring type Centring vectors Multiplicity (number of nodes per unit cell) Unit-cell volume Vc P Primitive 0 1 V A A-face centred ½bc+½cc 2 2V B B-face centred ½cc+½ac 2 2V C C-face centred ½ac+½bc 2 2V I body centred(Innenzentriert) ½ac+½bc+½cc 2 2V F All-face centred ½ac+½bc 4 4V ½bc+½cc ½cc+½ac R Primitive(rhombohedral axes) 0 1 V R Rhombohedrally centred(hexagonal axes) ⅔ac+⅓bc+⅓cc 3 3V ⅓ac+⅔bc+⅔cc H Hexagonally centred ⅔ac+⅓bc 3 3V ⅓ac+⅔bc