Reduced cell

(Difference between revisions)
 Revision as of 15:52, 18 December 2017 (view source) (see also + link + bold)← Older edit Revision as of 16:13, 18 December 2017 (view source) (geometrical meaning)Newer edit → Line 48: Line 48: *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 == == See also ==

Revision as of 16:13, 18 December 2017

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