Reduced cell

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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.

Contents

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).

See also