Reciprocal space - Revision history

BrianMcMahon: Rendering problem: replaced \bold with \mathbf

2021-02-03T15:22:00Z

Rendering problem: replaced \bold with \mathbf

BrianMcMahon: Tidied translations and corrected German and Spanish (U. Mueller)

2017-11-17T16:50:40Z

Tidied translations and corrected German and Spanish (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-16T15:59:55Z

Style edits to align with printed edition

MassimoNespolo: /* See also */ unordered list

2017-04-10T16:34:37Z

‎See also: unordered list

MassimoNespolo at 11:15, 26 February 2007

2007-02-26T11:15:17Z

GiovanniFerraris at 14:19, 28 March 2006

2006-03-28T14:19:29Z

UriShmueli at 15:23, 6 March 2006

2006-03-06T15:23:18Z

AndreAuthier at 09:47, 27 February 2006

2006-02-27T09:47:04Z

AndreAuthier at 09:22, 27 February 2006

2006-02-27T09:22:58Z

AndreAuthier at 09:21, 27 February 2006

2006-02-27T09:21:37Z

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 The reciprocal and direct spaces are reciprocal of one another, that is the reciprocal space associated to the reciprocal space is the direct space. They are related by a Fourier transform and the reciprocal space is also called ''Fourier space'' or ''phase space''.
-The '''vector product''' of two direct space vectors, <math>{\bold r_1} = u_1 {\bold a} + v_1 {\bold b} + w_1 {\bold c}</math> and <math>{\bold r_2} = u_2 {\bold a} + v_2 {\bold b} + w_2 {\bold c}</math>  is a reciprocal space vector,
+The '''vector product''' of two direct space vectors, <math>{\mathbf r_1} = u_1 {\mathbf a} + v_1 {\mathbf b} + w_1 {\mathbf c}</math> and <math>{\mathbf r_2} = u_2 {\mathbf a} + v_2 {\mathbf b} + w_2 {\mathbf c}</math>  is a reciprocal space vector,
 <center>
 <math>
-{\bold r*}  = {\bold r_1} \times {\bold r_2} = V (v_1 w_2 - v_2 w_1) {\bold a*} + V (w_1 u_2 - w_2 u_1) {\bold b*} + V (u_1 v_2 - u_2 v_1) {\bold c*}.
+{\mathbf r*}  = {\mathbf r_1} \times {\mathbf r_2} = V (v_1 w_2 - v_2 w_1) {\mathbf a*} + V (w_1 u_2 - w_2 u_1) {\mathbf b*} + V (u_1 v_2 - u_2 v_1) {\mathbf c*}.
 </math>
 </center>

← Older revision		Revision as of 16:50, 17 November 2017
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−	<~~Font~~ color="blue">Espace réciproque </~~Font~~>(''Fr''). <~~Font~~ color="red">~~Reziprokes~~ Raum </~~Font~~>(''Ge''). <~~Font~~ color="~~green~~">~~Espacio~~ reciproco </~~Font~~>(''Sp''). <~~Font~~ color="~~black~~"> ~~Spazio reciproco~~ </~~Font~~>(''It''). <~~Font~~ color="~~purple~~">~~逆空間~~</~~Font~~> (''Ja'').	+	<font color="blue">Espace réciproque</font> (''Fr''). <font color="red">Reziproker Raum</font> (''Ge''). <font color="black">Spazio reciproco</font> (''It''). <font color="purple">逆空間</font> (''Ja''). <font color="green">Espacio recíproco</font> (''Sp'').

@@ Line 1: / Line 1: @@
-<Font color="blue">Espace réciproque </Font>(''Fr''); <Font color="red">Reziprokes Raum </Font>(''Ge''); <Font color="green">Espacio reciproco </Font>(''Sp''); <Font color="black"> Spazio reciproco </Font>(''It''); <Font color="purple">逆空間</Font> (''Ja'').
+<Font color="blue">Espace réciproque </Font>(''Fr''). <Font color="red">Reziprokes Raum </Font>(''Ge''). <Font color="green">Espacio reciproco </Font>(''Sp''). <Font color="black"> Spazio reciproco </Font>(''It''). <Font color="purple">逆空間</Font> (''Ja'').
@@ Line 27: / Line 27: @@
 <center>
 <math>
-{\bold r*}  = {\bold r_1} \times {\bold r_2} = V (v_1 w_2 - v_2 w_1) {\bold a*} + V (w_1 u_2 - w_2 u_1) {\bold b*} + V (u_1 v_2 - u_2 v_1) {\bold c}.
+{\bold r*}  = {\bold r_1} \times {\bold r_2} = V (v_1 w_2 - v_2 w_1) {\bold a*} + V (w_1 u_2 - w_2 u_1) {\bold b*} + V (u_1 v_2 - u_2 v_1) {\bold c*}.
 </math>
 </center>
@@ Line 37: / Line 37: @@
 '''r''' . '''r*''' = ''uh'' + ''vk'' +''wl''.
 </center>
-In a '''change of coordinate system''', The coordinates of a vector in reciprocal space transform like the basis vectors in direct space and are called for that reason ''covariant''. The vectors in reciprocal transform like the coordinates in direct space and are called ''contravariant''.
+In a '''coordinate system change''', the coordinates of a vector in reciprocal space transform like the basis vectors in direct space and are called for that reason ''covariant''. The vectors in reciprocal space transform like the coordinates in direct space and are called ''contravariant''.
 == Geometrical relationships ==
-The '''volume''' ''V*'' = ('''a*''', '''b*''', '''c*''') of the cell constructed on the reciprocal vectors '''a*''','''b*''' and '''c*''' is equal to 1/''V''.
+The '''volume''' ''V*'' = ('''a*''', '''b*''', '''c*''') of the cell constructed on the reciprocal vectors '''a*''', '''b*''' and '''c*''' is equal to 1/''V''.
 The '''lengths''' ''a*'', ''b*'', ''c*'' of the reciprocal basis vectors and the '''angles''', &#945;*, &#946;*, &#947;*, between the pairs of reciprocal vectors ('''b*''', '''c*'''), ('''c*''', '''a*'''), ('''a*''', '''b*'''), are related to the corresponding lengths and angles for the direct basis vectors through the following relations:
@@ Line 47: / Line 47: @@
 ''a*'' = ''b'' ''c'' sin &#945;/''V''; ''b*'' = ''c'' ''a'' sin &#946;/''V''; ''c*'' = ''a'' ''b'' sin &#947;/''V'';
-cos &#945;* = (cos &#946;cos &#947; - cos &#945;)/|sin &#946; sin &#947;|;
+cos &#945;* = (cos &#946;cos &#947; - cos &#945;)/(sin &#946; sin &#947;);<br>
-cos &#946;* = (cos &#947;cos &#945; - cos &#946;)/|sin &#947; sin &#945;|;
+cos &#946;* = (cos &#947;cos &#945; - cos &#946;)/(sin &#947; sin &#945;);<br>
-cos &#947;* = (cos &#945;cos &#946; - cos &#947;)/|sin &#945; sin &#945;|.
+cos &#947;* = (cos &#945;cos &#946; - cos &#947;)/(sin &#945; sin &#946;).
 == History ==
-The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs (1881 - ''Elements of Vector Analysis, arranged for the Use of Students in Physics''. Yale University, New Haven).
+The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs [(1881). ''Elements of Vector Analysis, arranged for the Use of Students in Physics''. Yale University, New Haven].
 == See also ==
-*[[reciprocal lattice]]
+*[[Reciprocal lattice]]
-*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')
+*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ ''The Reciprocal Lattice'']  (Teaching Pamphlet No. 4 of the International Union of Crystallography)
-*Section 1.1, ''International Tables of Crystallography, Volume B''
+*Chapter 1.1 of ''International Tables for Crystallography, Volume B''
-*Section 1.1, ''International Tables of Crystallography, Volume C''
+*Chapter 1.1 of ''International Tables for Crystallography, Volume C''
-*Section 1.1.2, ''International Tables of Crystallography, Volume D''
+*Chapter 1.1.2 of ''International Tables for Crystallography, Volume D''
 [[Category:Fundamental crystallography]]

@@ Line 57: / Line 57: @@
 == See also ==
+*[[reciprocal lattice]]
-[[reciprocal lattice]]
+*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')
+*Section 1.1, ''International Tables of Crystallography, Volume B''
-[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')
+*Section 1.1, ''International Tables of Crystallography, Volume C''
+*Section 1.1.2, ''International Tables of Crystallography, Volume D''
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 11:15, 26 February 2007
Line 1:		Line 1:
−	<Font color="blue">Espace réciproque </Font>(''Fr''). <Font color="red">Reziprokes Raum </Font>(''Ge''). <Font color="green">Espacio reciproco </Font>(''Sp''). <Font color="black"> Spazio reciproco </Font>(''It'')	+	<Font color="blue">Espace réciproque </Font>(''Fr''); <Font color="red">Reziprokes Raum </Font>(''Ge''); <Font color="green">Espacio reciproco </Font>(''Sp''); <Font color="black"> Spazio reciproco </Font>(''It''); <Font color="purple">逆空間</Font> (''Ja'').

← Older revision		Revision as of 14:19, 28 March 2006
Line 1:		Line 1:
−	<Font color="blue">Espace réciproque </Font>(''Fr''). <Font color="red">Reziprokes Raum </Font>(''Ge''). <Font color="green">Espacio reciproco </Font>(''Sp'').	+	<Font color="blue">Espace réciproque </Font>(''Fr''). <Font color="red">Reziprokes Raum </Font>(''Ge''). <Font color="green">Espacio reciproco </Font>(''Sp''). <Font color="black"> Spazio reciproco </Font>(''It'')
		+

	== Definition ==		== Definition ==

← Older revision		Revision as of 09:47, 27 February 2006
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−	~~= reciprocal space =~~
−
	<Font color="blue">Espace réciproque </Font>(''Fr''). <Font color="red">Reziprokes Raum </Font>(''Ge''). <Font color="green">Espacio reciproco </Font>(''Sp'').		<Font color="blue">Espace réciproque </Font>(''Fr''). <Font color="red">Reziprokes Raum </Font>(''Ge''). <Font color="green">Espacio reciproco </Font>(''Sp'').

@@ Line 1: / Line 1: @@
 = reciprocal space =
 <Font color="blue">Espace réciproque </Font>(''Fr''). <Font color="red">Reziprokes Raum </Font>(''Ge''). <Font color="green">Espacio reciproco </Font>(''Sp'').
 == Definition ==
@@ Line 57: / Line 53: @@
-==== History ====
+== History ==
 The notion of reciprocal vectors was introduced in vector analysis by J. W. Gibbs (1981 - ''Elements of Vector Analysis, arranged for the Use of Students in Physics''. Yale University, New Haven).
-=== See also ===
+== See also ==
 [[reciprocal lattice]]