Reciprocal lattice - Revision history

BrianMcMahon: Tidied translations.

2017-12-15T10:36:24Z

Tidied translations.

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

2017-11-17T16:48:15Z

Tidied translations and corrected Spanish (U. Mueller)

BrianMcMahon at 15:14, 16 May 2017

2017-05-16T15:14:31Z

BrianMcMahon: Style edits to align with printed edition

2017-05-16T15:13:50Z

Style edits to align with printed edition

BrianMcMahon: Style edits to align with printed edition

2017-05-16T15:05:29Z

Style edits to align with printed edition

MassimoNespolo: /* See also */ 6th edition of ITA

2017-04-10T16:33:37Z

‎See also: 6th edition of ITA

BrianMcMahon at 10:18, 25 May 2007

2007-05-25T10:18:26Z

MassimoNespolo: /* See also */

2007-04-09T10:27:21Z

‎See also

MassimoNespolo: /* See also */ link

2007-04-09T10:10:37Z

‎See also: link

MassimoNespolo at 11:14, 26 February 2007

2007-02-26T11:14:38Z

@@ Line 1: / Line 1: @@
-<font color="blue">Réseau réciproque</font> (''Fr''). <font color="red">Reziprokes Gitter</font> (''Ge''). <font color="black">Reticolo reciproco </font>(''It''). <font color="purple">逆格子</font> (''Ja''). <font color="green">Red recíproca</font> (''Sp'').
+<font color="blue">Réseau réciproque</font> (''Fr''). <font color="red">Reziprokes Gitter</font> (''Ge''). <font color="black">Reticolo reciproco</font> (''It''). <font color="purple">逆格子</font> (''Ja''). <font color="green">Red recíproca</font> (''Sp'').
@@ Line 40: / Line 40: @@
 ''w''/(''h<sub>1</sub>'' ''k<sub>2</sub>'' - ''h<sub>2</sub>'' ''k<sub>1</sub>'').
 </center>
 == [[Centred lattices]] ==

← Older revision		Revision as of 16:48, 17 November 2017
Line 1:		Line 1:
−	<~~Font~~ color="blue">Réseau réciproque </~~Font~~>(''Fr''). <~~Font~~ color="red">Reziprokes Gitter (''Ge'')~~</Font>~~. <~~Font~~ color="~~green~~">~~Red reciproca~~ </~~Font~~>(''Sp''). <~~Font~~ color="~~black~~">~~Reticolo reciproco~~ </~~Font~~>(''It''). <~~Font~~ color="~~purple~~">~~逆格子~~</~~Font~~> (''Ja'').	+	<font color="blue">Réseau réciproque</font> (''Fr''). <font color="red">Reziprokes Gitter</font> (''Ge''). <font color="black">Reticolo reciproco </font>(''It''). <font color="purple">逆格子</font> (''Ja''). <font color="green">Red recíproca</font> (''Sp'').

@@ Line 187: / Line 187: @@
 *[[Reciprocal space]]
 *[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet No. 4 of the International Union of Crystallography)
-*Chapter 1.3.2.5, ''International Tables for Crystallography, Volume A,'' 6th edition
+*Chapter 1.3.2.5 of ''International Tables for Crystallography, Volume A,'' 6th edition
-*Chapter 1.1, ''International Tables for Crystallography, Volume B''
+*Chapter 1.1 of ''International Tables for Crystallography, Volume B''
-*Chapter 1.1, ''International Tables for Crystallography, Volume C''
+*Chapter 1.1 of ''International Tables for Crystallography, Volume C''
-*Chapter 1.1.2, ''International Tables for Crystallography, Volume D''
+*Chapter 1.1.2 of ''International Tables for Crystallography, Volume D''
 [[Category:Fundamental crystallography]]

@@ Line 141: / Line 141: @@
 where '''a<sub>c</sub>''', '''b<sub>c</sub>''', '''c<sub>c</sub>''' are the basis vectors of the conventional multiple cell and '''a*<sub>c</sub>''', '''b*<sub>c</sub>''', '''c*<sub>c</sub>''' the corresponding reciprocal lattice vectors.
-An elementary proof that the reciprocal lattice of a face-centred lattice ''F'' is a body-centred lattice ''I'' and, reciprocally, is given in [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice].
+An elementary proof that the reciprocal lattice of a face-centred lattice ''F'' is a body-centred lattice ''I'' and, reciprocally, is given in [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ ''The Reciprocal Lattice''] (Teaching Pamphlet No. 4 of the International Union of Crystallography).
 == Diffraction condition in reciprocal space ==
@@ Line 147: / Line 147: @@
 [[Image:ReciprocalLattice-1.gif|right]]
 The condition that the waves outgoing from two point scatterers separated by a lattice vector
-'''r''' = ''u'' '''a''' + ''v'' '''b''' + ''w'' '''c''' (''u'', ''v'', ''w'' integers) be in phase is that the scalar product ('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) . '''r''', where '''s<sub>h</sub>''' and '''s<sub>o</sub>''' are unit vectors in the scattered and incident directions, respectively, be an integer, ''n''. This condition is satisfied whatever '''r''' if the diffraction vector ('''OH''' = '''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) is of the form:
+'''r''' = ''u'' '''a''' + ''v'' '''b''' + ''w'' '''c''' (''u'', ''v'', ''w'' integers) be in phase is that the scalar product ('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) &sdot; '''r''', where '''s<sub>h</sub>''' and '''s<sub>o</sub>''' are unit vectors in the scattered and incident directions, respectively, be an integer, ''n''. This condition is satisfied whatever '''r''' if the diffraction vector ('''OH''' = '''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) is of the form:
 <center>
 ('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''',
@@ Line 153: / Line 153: @@
 where ''h'', ''k'', ''l'' are integers, namely the diffraction vector '''OH''' is a vector of the reciprocal lattice (Fig. 1).
-A node of the reciprocal lattice is therefore associated to each Bragg reflection on the lattice
+A node of the reciprocal lattice is therefore associated with each Bragg reflection on the lattice
 planes of [[Miller indices]] (''h/K'', ''k/K'', ''l/K''). It is called the ''hkl'' reflection.
-The relation '''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955; =  '''0H''' generalizes the [[Laue equations]]. It is equivalent to [[Bragg's law]], as can be seen in Fig. 2.
+The relation '''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955; =  '''OH''' generalizes the [[Laue equations]]. It is equivalent to [[Bragg's law]], as can be seen in Fig. 2.
 [[Image:ReciprocalLattice-2.gif]]
@@ Line 178: / Line 178: @@
 == 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; reprinted: Gibbs J. W. and Wilson E. B., 1902, ''Vector analysis'', New York; 1960, Dover Publications). The concept of reciprocal lattice was adapted by P. P. Ewald to interpret the diffraction pattern of an orthorhombic crystal (1913) in his famous paper where he introduced the sphere of diffraction. It was extended to lattices of any type of symmetry by M. von Laue
+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; reprinted: Gibbs, J. W. and Wilson, E. B. (1902). ''Vector analysis'', New York; 1960, Dover Publications]. The concept of reciprocal lattice was adapted by P. P. Ewald to interpret the diffraction pattern of an orthorhombic crystal (1913) in his famous paper where he introduced the sphere of diffraction. It was extended to lattices of any type of symmetry by M. von Laue (1914) and Ewald (1921). The first approach to that concept is that of the system of
-(1914) and Ewald (1921). The first approach to that concept is that of the system of
+''polar axes'', introduced by Bravais in 1850, which associates the direction of its normal with a family of lattice planes.
 == See also ==
@@ Line 187: / Line 186: @@
 *[[Polar lattice]]
 *[[Reciprocal space]]
-*[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 1.3.2.5, ''[http://it.iucr.org/A/ International Tables of Crystallography, Volume A]''
+*Chapter 1.3.2.5, ''International Tables for Crystallography, Volume A,'' 6th edition
-*Section 1.1, ''[http://it.iucr.org/B/ International Tables of Crystallography, Volume B]''
+*Chapter 1.1, ''International Tables for Crystallography, Volume B''
-*Section 1.1, ''[http://it.iucr.org/C/ International Tables of Crystallography, Volume C]''
+*Chapter 1.1, ''International Tables for Crystallography, Volume C''
-*Section 1.1.2, ''[http://it.iucr.org/D/ International Tables of Crystallography, Volume D]''
+*Chapter 1.1.2, ''International Tables for Crystallography, Volume D''
 [[Category:Fundamental crystallography]]

@@ Line 1: / Line 1: @@
-<Font color="blue">Réseau réciproque </Font>(''Fr''); <Font color="red">Reziprokes Gitter (''Ge'')</Font>; <Font color="green">Red reciproca </Font>(''Sp''); <Font color="black"> Reticolo reciproco </Font>(''It''); <Font color="purple">逆格子</Font> (''Ja'').
+<Font color="blue">Réseau réciproque </Font>(''Fr''). <Font color="red">Reziprokes Gitter (''Ge'')</Font>. <Font color="green">Red reciproca </Font>(''Sp''). <Font color="black">Reticolo reciproco </Font>(''It''). <Font color="purple">逆格子</Font> (''Ja'').
 == Definition ==
-The reciprocal lattice is constituted by the set of all possible linear combinations of the basis vectors
+The reciprocal lattice is constituted of the set of all possible linear combinations of the basis vectors
 '''a*''', '''b*''', '''c*''' of the [[reciprocal space]]. A point (''node''), ''H'', of the reciprocal lattice is defined by its position vector:
 <center>
-'''OH''' = '''r<sub>hkl</sub>*''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*'''.
+'''OH''' = '''r*<sub>hkl</sub>''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*'''.
 </center>
 If ''H'' is the ''n''th node on the row ''OH'', one has:
@@ Line 14: / Line 14: @@
 '''OH''' = ''n'' '''OH<sub>1</sub>''' = ''n'' (''h''<sub>1</sub> '''a*''' + ''k''<sub>1</sub> '''b*''' + ''l''<sub>1</sub> '''c*'''),
 </center>
-where ''H''<sub>1</sub> is the first node on the row ''OH'' and ''h''<sub>1</sub> , ''k''<sub>1</sub> , ''l''<sub>1</sub> are relatively prime.
+where ''H''<sub>1</sub> is the first node on the row ''OH'' and ''h''<sub>1</sub>, ''k''<sub>1</sub>, ''l''<sub>1</sub> are relatively prime.
-The generalizaion of the reciprocal lattice in a four-dimensional space for incommensurate structures is described in Section 9.8 of ''International Tables of Crystallography, Volume C''.
+The generalization of the reciprocal lattice in a four-dimensional space for incommensurate structures is described in Chapter 9.8 of ''International Tables for Crystallography, Volume C''.
 == Geometrical applications ==
-Each '''vector''' '''OH''' = '''r<sub>hkl</sub>*''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''' ''' of the reciprocal lattice is associated to a family of direct lattice planes'''. It is normal to the planes of the family, and the lattice spacing of the family is ''d'' = 1/''OH''<sub>1</sub> = ''n''/''OH'' if ''H'' is the ''n''th node on the reciprocal lattice row ''OH''. One usually sets ''d<sub>hkl</sub>'' = ''d''/''n'' = 1/''OH''. If '''OP''' = ''x'' '''a''' + ''y'' '''b''' + ''z'' '''c''' is the position vector of a point of a lattice plane, the equation of the plane is given by '''OH'''<sub>1</sub> . '''OP''' = ''K'' where ''K'' is a constant integer. Using the properties of the scalar product of a [[reciprocal space]] vector and a [[direct space]] vector, this equation is
+Each '''vector''' '''OH''' = '''r*<sub>hkl</sub>''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''' ''' of the reciprocal lattice is associated with a family of direct lattice planes'''. It is normal to the planes of the family, and the lattice spacing of the family is ''d'' = 1/''OH''<sub>1</sub> = ''n''/''OH'' if ''H'' is the ''n''th node on the reciprocal lattice row ''OH''. One usually sets ''d<sub>hkl</sub>'' = ''d''/''n'' = 1/''OH''. If '''OP''' = ''x'' '''a''' + ''y'' '''b''' + ''z'' '''c''' is the position vector of a point of a lattice plane, the equation of the plane is given by '''OH'''<sub>1</sub> &sdot; '''OP''' = ''K'' where ''K'' is a constant integer. Using the properties of the scalar product of a [[reciprocal space]] vector and a [[direct space]] vector, this equation is
-'''OH'''<sub>1</sub> . '''OP''' = ''h<sub>1</sub>''x'' + ''k''<sub>1</sub>''y'' + ''l''<sub>1</sub>z'' = ''K''. The [[Miller indices]] of the family are
+'''OH'''<sub>1</sub> &sdot; '''OP''' = ''h<sub>1</sub>''x'' + ''k''<sub>1</sub>''y'' + ''l''<sub>1</sub>z'' = ''K''. The [[Miller indices]] of the family are
 ''h''<sub>1</sub>, ''k''<sub>1</sub>, ''l''<sub>1</sub>. The subscripts of the Miller indices will be dropped hereafter.
@@ Line 34: / Line 34: @@
 </center>
 The coordinates ''u'', ''v'', ''w'' in direct space of the '''[[zone axis]] intersection of two families of lattice planes''' of Miller indices ''h<sub>1</sub>'', ''k<sub>1</sub>'', ''l<sub>1</sub>'' and
-''h<sub>2</sub>'', ''k<sub>2</sub>'', ''l<sub>2</sub>'', respectively, are proportional to the coordinates of the vector product of the reciprocal lattice vectors associated to these two families:
+''h<sub>2</sub>'', ''k<sub>2</sub>'', ''l<sub>2</sub>'', respectively, are proportional to the coordinates of the vector product of the reciprocal lattice vectors associated with these two families:
 <center>
 ''u''/(''k<sub>1</sub>'' ''l<sub>2</sub>'' - ''k<sub>2</sub>'' ''l<sub>1</sub>'') =

← Older revision		Revision as of 11:14, 26 February 2007
Line 1:		Line 1:
−	<Font color="blue">Réseau réciproque </Font>(''Fr''). <Font color="red">Reziprokes Gitter (''Ge'')</Font>. <Font color="green">Red reciproca </Font>(''Sp'')<Font color="black"> Reticolo reciproco </Font>(''It'')	+	<Font color="blue">Réseau réciproque </Font>(''Fr''); <Font color="red">Reziprokes Gitter (''Ge'')</Font>; <Font color="green">Red reciproca </Font>(''Sp''); <Font color="black"> Reticolo reciproco </Font>(''It''); <Font color="purple">逆格子</Font> (''Ja'').