Reduced cell - Revision history

MassimoNespolo: No, it is cell（単位胞）not lattice (格子)

2019-03-27T08:54:33Z

No, it is cell（単位胞）not lattice (格子)

ShigeruOhba at 13:40, 26 March 2019

2019-03-26T13:40:14Z

MassimoNespolo: geometrical meaning

2017-12-18T16:13:29Z

geometrical meaning

MassimoNespolo: see also + link + bold

2017-12-18T15:52:04Z

MassimoNespolo: Languages

2017-12-18T15:48:29Z

Languages

MassimoNespolo: Created page with "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]..."

2017-12-18T15:43:25Z

Created page with "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]..."

New page

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

<math>T = (\mathbf{a}\cdot\mathbf{b})(\mathbf{b}\cdot\mathbf{c})(\mathbf{c}\cdot\mathbf{a})</math>.

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===
*<math>\mathbf{a}\cdot\mathbf{a}</math> ≤ <math>\mathbf{b}\cdot\mathbf{b}</math> ≤ <math>\mathbf{c}\cdot\mathbf{c}</math>
*<math>|\mathbf{b}\cdot\mathbf{c}|</math> ≤ <math>(\mathbf{b}\cdot\mathbf{b})/2</math>
*<math>|\mathbf{a}\cdot\mathbf{c}|</math> ≤ <math>(\mathbf{a}\cdot\mathbf{a})/2</math>
*<math>|\mathbf{a}\cdot\mathbf{b}|</math> ≤ <math>(\mathbf{a}\cdot\mathbf{a})/2</math>
*<math>\mathbf{b}\cdot\mathbf{c} > 0</math>
*<math>\mathbf{a}\cdot\mathbf{c} > 0</math>
*<math>\mathbf{a}\cdot\mathbf{b} > 0</math>

===Special conditions===
*if <math>\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b}</math> then <math>\mathbf{b}\cdot\mathbf{c}</math> ≤ <math>\mathbf{a}\cdot\mathbf{c}</math>
*if <math>\mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c}</math> then <math>\mathbf{a}\cdot\mathbf{c}</math> ≤ <math>\mathbf{a}\cdot\mathbf{b}</math>
*if <math>\mathbf{b}\cdot\mathbf{c} = (\mathbf{b}\cdot\mathbf{b})</math>/2 then <math>\mathbf{a}\cdot\mathbf{b}</math> ≤ <math>2\mathbf{a}\cdot\mathbf{c}</math>
*if <math> \mathbf{a}\cdot\mathbf{c} = (\mathbf{a}\cdot\mathbf{a})</math>/2 then <math>\mathbf{a}\cdot\mathbf{b}</math> ≤ <math>2\mathbf{b}\cdot\mathbf{c}</math>
*if <math> \mathbf{a}\cdot\mathbf{b} = (\mathbf{a}\cdot\mathbf{a})</math>/2 then <math>\mathbf{a}\cdot\mathbf{c}</math> ≤ <math>2\mathbf{b}\cdot\mathbf{c}</math>

==Type-II cell==
===Main conditions===
*<math>\mathbf{a}\cdot\mathbf{a}</math> ≤ <math>\mathbf{b}\cdot\mathbf{b}</math> ≤ <math>\mathbf{c}\cdot\mathbf{c}</math>
*<math>|\mathbf{b}\cdot\mathbf{c}|</math> ≤ <math>(\mathbf{b}\cdot\mathbf{b})/2</math>
*<math>|\mathbf{a}\cdot\mathbf{c}|</math> ≤ <math>(\mathbf{a}\cdot\mathbf{a})/2</math>
*<math>|\mathbf{a}\cdot\mathbf{b}|</math> ≤ <math>(\mathbf{a}\cdot\mathbf{a})/2</math>
*<math>(|\mathbf{b}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{c}|+|\mathbf{a}\cdot\mathbf{b}|)</math> ≤ <math>(\mathbf{a}\cdot\mathbf{a}+\mathbf{b}\cdot\mathbf{b})/2</math>
*<math>\mathbf{b}\cdot\mathbf{c}</math> ≤ 0
*<math>\mathbf{a}\cdot\mathbf{c}</math> ≤ 0
*<math>\mathbf{a}\cdot\mathbf{b}</math> ≤ 0

===Special conditions===
*if <math>\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b}</math> then <math>|\mathbf{b}\cdot\mathbf{c}|</math> ≤ <math>|\mathbf{a}\cdot\mathbf{c}|</math>
*if <math>\mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c}</math> then <math>|\mathbf{a}\cdot\mathbf{c}|</math> ≤ <math>|\mathbf{a}\cdot\mathbf{b}|</math>
*if <math>|\mathbf{b}\cdot\mathbf{c}| = (\mathbf{b}\cdot\mathbf{b})/2</math> then <math>\mathbf{a}\cdot\mathbf{b} = 0</math>
*if <math>|\mathbf{a}\cdot\mathbf{c}| = (\mathbf{a}\cdot\mathbf{a})</math>/2 then <math>\mathbf{a}\cdot\mathbf{b} = 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> ≤ <math>2|\mathbf{a}\cdot\mathbf{c}|+ |\mathbf{a}\cdot\mathbf{b}|</math>

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 08:54, 27 March 2019
Line 1:		Line 1:
−	<font color="blue">Maille réduite</font> (''Fr''). <font color="black">Cella ridotta</font> (''It''). <font color="purple">~~既約格子~~</font> (''Ja'').	+	<font color="blue">Maille réduite</font> (''Fr''). <font color="black">Cella ridotta</font> (''It''). <font color="purple">既約単位胞</font> (''Ja'').

← Older revision		Revision as of 13:40, 26 March 2019
Line 1:		Line 1:
−	<font color="blue">Maille réduite</font> (''Fr''). <font color="black">Cella ridotta</font> (''It''). <font color="purple">~~規約単位胞~~</font> (''Ja'').	+	<font color="blue">Maille réduite</font> (''Fr''). <font color="black">Cella ridotta</font> (''It''). <font color="purple">既約格子</font> (''Ja'').

← Older revision		Revision as of 16:13, 18 December 2017
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	*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> ≤ <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> ≤ <math>2\|\mathbf{a}\cdot\mathbf{c}\|+ \|\mathbf{a}\cdot\mathbf{b}\|</math>

		+	==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 ==

@@ Line 2: / Line 2: @@
-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
@@ Line 47: / Line 47: @@
 *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>
 [[Category:Fundamental crystallography]]

@@ Line 1: / Line 1: @@
 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
@@ Line 6: / Line 9: @@
 If ''T'' > 0, the cell is of type I, if ''T'' &#8804; 0 it is of type II.
-The conditions for a primitive cell to be a reduced cell can all be stated analytically as follows:
+The conditions for a primitive cell to be a reduced cell can all be stated analytically as follows.
 ==Type-I cell==