# Reduced cell

(Difference between revisions)
 Revision as of 15:48, 18 December 2017 (view source) (Languages)← Older edit Latest revision as of 08:54, 27 March 2019 (view source) (No, it is cell（単位胞）not lattice (格子)) (3 intermediate revisions not shown) Line 1: Line 1: - Maille réduite (''Fr''). Cella ridotta (''It''). 規約単位胞 (''Ja''). + Maille réduite (''Fr''). Cella ridotta (''It''). 既約単位胞 (''Ja''). - A primitive basis a, b, c is called a ‘reduced basis’ if it is right-handed and if the components of the [[metric tensor]] satisfy the conditions below. Because of [[lattice]] [[symmetry operation|symmetry]] there can be two or more possible orientations of the reduced basis in a given lattice but, apart from orientation, the reduced basis is unique. + A [[primitive basis]] '''a''', '''b''', '''c''' is called a '''reduced basis''' if it is right-handed and if the components of the [[metric tensor]] satisfy the conditions below. Because of [[lattice]] [[symmetry operation|symmetry]] there can be two or more possible orientations of the reduced basis in a given lattice but, apart from orientation, the reduced basis is unique. The type of a cell depends on the sign of The type of a cell depends on the sign of Line 47: Line 47: *if  $\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{c} = 0$ *if  $\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{c} = 0$ *if $(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|) = (\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2$ then  $\mathbf{a}\cdot\mathbf{a}$ ≤ $2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|$ *if $(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|) = (\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2$ then  $\mathbf{a}\cdot\mathbf{a}$ ≤ $2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|$ + + ==Geometrical meaning of the reduced cell== + The main conditions express the following two requirements: + *Of all lattice vectors, none is shorter than '''a'''; of those not directed along '''a''', none is shorter than '''b'''; of those not lying in the '''ab''' plane, none is shorter than '''c'''. + *The three angles between basis vectors are either all acute (type I) or all non-acute (type II). + + == See also == + *[[Conventional cell]] + *[[Crystallographic basis]] + *[[Direct lattice]] + *[[Unit cell]] + *Chapter 3.1.3. of ''International Tables for Crystallography, Volume A'', 6th edition [[Category:Fundamental crystallography]] [[Category:Fundamental crystallography]]

## Latest revision as of 08:54, 27 March 2019

Maille réduite (Fr). Cella ridotta (It). 既約単位胞 (Ja).

A primitive basis a, b, c is called a reduced basis if it is right-handed and if the components of the metric tensor satisfy the conditions below. Because of lattice symmetry there can be two or more possible orientations of the reduced basis in a given lattice but, apart from orientation, the reduced basis is unique. The type of a cell depends on the sign of $T = (\mathbf{a}\cdot\mathbf{b})(\mathbf{b}\cdot\mathbf{c})(\mathbf{c}\cdot\mathbf{a})$.

If T > 0, the cell is of type I, if T ≤ 0 it is of type II.

The conditions for a primitive cell to be a reduced cell can all be stated analytically as follows.

## Type-I cell

### Main conditions

• $\mathbf{a}\cdot\mathbf{a}$ $\mathbf{b}\cdot\mathbf{b}$ $\mathbf{c}\cdot\mathbf{c}$
• $|\mathbf{b}\cdot\mathbf{c}|$ $(\mathbf{b}\cdot\mathbf{b})/2$
• $|\mathbf{a}\cdot\mathbf{c}|$ $(\mathbf{a}\cdot\mathbf{a})/2$
• $|\mathbf{a}\cdot\mathbf{b}|$ $(\mathbf{a}\cdot\mathbf{a})/2$
• $\mathbf{b}\cdot\mathbf{c} > 0$
• $\mathbf{a}\cdot\mathbf{c} > 0$
• $\mathbf{a}\cdot\mathbf{b} > 0$

### Special conditions

• if $\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b}$ then $\mathbf{b}\cdot\mathbf{c}$ $\mathbf{a}\cdot\mathbf{c}$
• if $\mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c}$ then $\mathbf{a}\cdot\mathbf{c}$ $\mathbf{a}\cdot\mathbf{b}$
• if $\mathbf{b}\cdot\mathbf{c} = (\mathbf{b}\cdot\mathbf{b})$/2 then $\mathbf{a}\cdot\mathbf{b}$ $2\mathbf{a}\cdot\mathbf{c}$
• if $\mathbf{a}\cdot\mathbf{c} = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{b}$ $2\mathbf{b}\cdot\mathbf{c}$
• if $\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{c}$ $2\mathbf{b}\cdot\mathbf{c}$

## Type-II cell

### Main conditions

• $\mathbf{a}\cdot\mathbf{a}$ $\mathbf{b}\cdot\mathbf{b}$ $\mathbf{c}\cdot\mathbf{c}$
• $|\mathbf{b}\cdot\mathbf{c}|$ $(\mathbf{b}\cdot\mathbf{b})/2$
• $|\mathbf{a}\cdot\mathbf{c}|$ $(\mathbf{a}\cdot\mathbf{a})/2$
• $|\mathbf{a}\cdot\mathbf{b}|$ $(\mathbf{a}\cdot\mathbf{a})/2$
• $(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|)$ $(\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2$
• $\mathbf{b}\cdot\mathbf{c}$ ≤ 0
• $\mathbf{a}\cdot\mathbf{c}$ ≤ 0
• $\mathbf{a}\cdot\mathbf{b}$ ≤ 0

### Special conditions

• if $\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b}$ then $|\mathbf{b}\cdot\mathbf{c}|$ $|\mathbf{a}\cdot\mathbf{c}|$
• if $\mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c}$ then $|\mathbf{a}\cdot\mathbf{c}|$ $|\mathbf{a}\cdot\mathbf{b}|$
• if $|\mathbf{b}\cdot\mathbf{c}| = (\mathbf{b}\cdot\mathbf{b})/2$ then $\mathbf{a}\cdot\mathbf{b} = 0$
• if $|\mathbf{a}\cdot\mathbf{c}| = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{b} = 0$
• if $\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})$/2 then $\mathbf{a}\cdot\mathbf{c} = 0$
• if $(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|) = (\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2$ then $\mathbf{a}\cdot\mathbf{a}$ $2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|$

## Geometrical meaning of the reduced cell

The main conditions express the following two requirements:

• Of all lattice vectors, none is shorter than a; of those not directed along a, none is shorter than b; of those not lying in the ab plane, none is shorter than c.
• The three angles between basis vectors are either all acute (type I) or all non-acute (type II).