Reduced cell

From Online Dictionary of Crystallography

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(No, it is cell(単位胞)not lattice (格子))
 
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<font color="blue">Maille réduite</font> (''Fr''). <font color="black">Cella ridotta</font> (''It''). <font color="purple">規約単位胞</font> (''Ja'').
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<font color="blue">Maille réduite</font> (''Fr''). <font color="black">Cella ridotta</font> (''It''). <font color="purple">既約単位胞</font> (''Ja'').
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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.
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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
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*if  <math>\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})</math>/2 then <math>\mathbf{a}\cdot\mathbf{c} = 0</math>
*if  <math>\mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})</math>/2 then <math>\mathbf{a}\cdot\mathbf{c} = 0</math>
*if <math>(|\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</math> then  <math>\mathbf{a}\cdot\mathbf{a}</math> &#8804; <math>2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|</math>
*if <math>(|\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</math> then  <math>\mathbf{a}\cdot\mathbf{a}</math> &#8804; <math>2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|</math>
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==Geometrical meaning of the reduced cell==
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The main conditions express the following two requirements:
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*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'''.
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*The three angles between basis vectors are either all acute (type I) or all non-acute (type II).
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== See also ==
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*[[Conventional cell]]
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*[[Crystallographic basis]]
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*[[Direct lattice]]
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*[[Unit cell]]
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*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.

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