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Difference between revisions of "Reciprocal lattice"

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= reciprocal lattice =
+
<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'').
  
 
=== Other languages ===
 
 
Réseau réciproque (''Fr''). Reziprokes Gitter (''Ge''). Red reciproca (''Sp'')
 
  
 
== Definition ==
 
== 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:  
 
'''a*''', '''b*''', '''c*''' of the [[reciprocal space]]. A point (''node''), ''H'', of the reciprocal lattice is defined by its position vector:  
  
'''OH''' = '''r<sub>hkl</sub>*''' = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*'''.
+
<center>
 
+
'''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:
 
If ''H'' is the ''n''th node on the row ''OH'', one has:
 
+
<center>
 
'''OH''' = ''n'' '''OH<sub>1</sub>''' = ''n'' (''h''<sub>1</sub> '''a*''' + ''k''<sub>1</sub> '''b*''' + ''l''<sub>1</sub> '''c*'''),
 
'''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 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''.
 
 
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''.
 
  
 
== Geometrical applications ==
 
== 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''' = ''h22<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.
 
''h''<sub>1</sub>, ''k''<sub>1</sub>, ''l''<sub>1</sub>. The subscripts of the Miller indices will be dropped hereafter.
  
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'''r<sub>2</sub>''' = ''u<sub>2</sub>'' '''a''' + ''v<sub>2</sub>'' '''b''' + ''w<sub>2</sub>'' '''c''' are proportional to the
 
'''r<sub>2</sub>''' = ''u<sub>2</sub>'' '''a''' + ''v<sub>2</sub>'' '''b''' + ''w<sub>2</sub>'' '''c''' are proportional to the
 
coordinates in reciprocal space, ''h'', ''k'', ''l'', of the vector product of these two vectors:
 
coordinates in reciprocal space, ''h'', ''k'', ''l'', of the vector product of these two vectors:
 
+
<center>
 
''h''/(''v<sub>1</sub>'' ''w<sub>2</sub>'' - ''v<sub>2</sub>'' ''w<sub>1</sub>'') =
 
''h''/(''v<sub>1</sub>'' ''w<sub>2</sub>'' - ''v<sub>2</sub>'' ''w<sub>1</sub>'') =
 
''k''/(''w<sub>1</sub>'' ''u<sub>2</sub>'' - ''w<sub>2</sub>'' ''u<sub>1</sub>'') =
 
''k''/(''w<sub>1</sub>'' ''u<sub>2</sub>'' - ''w<sub>2</sub>'' ''u<sub>1</sub>'') =
 
''l''/(''u<sub>1</sub>'' ''v<sub>2</sub>'' - ''u<sub>2</sub>'' ''v<sub>1</sub>'').
 
''l''/(''u<sub>1</sub>'' ''v<sub>2</sub>'' - ''u<sub>2</sub>'' ''v<sub>1</sub>'').
 
+
</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  
+
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>'') =
 
''u''/(''k<sub>1</sub>'' ''l<sub>2</sub>'' - ''k<sub>2</sub>'' ''l<sub>1</sub>'') =
 
''v''/(''l<sub>1</sub>'' ''h<sub>2</sub>'' - ''l<sub>2</sub>'' ''h<sub>1</sub>'') =
 
''v''/(''l<sub>1</sub>'' ''h<sub>2</sub>'' - ''l<sub>2</sub>'' ''h<sub>1</sub>'') =
 
''w''/(''h<sub>1</sub>'' ''k<sub>2</sub>'' - ''h<sub>2</sub>'' ''k<sub>1</sub>'').
 
''w''/(''h<sub>1</sub>'' ''k<sub>2</sub>'' - ''h<sub>2</sub>'' ''k<sub>1</sub>'').
 +
</center>
 +
 +
== [[Centred lattice|Centred lattices]] ==
 +
  
== Centred lattices ==
+
<table border cellspacing=0 cellpadding=5 align=center>
 +
  <tr>
 +
  <th colspan =3 align=center> Direct lattice</th>
 +
  <th colspan =3 align=center> Reciprocal lattice</th>
 +
  </tr>
 +
  <tr align=center>
 +
    <td>Bravais letter</td>
 +
    <td>Centring vectors</td>
 +
    <td>Unit-cell volume ''V<sub>c</sub>''</td>
 +
    <td>Bravais letter</td>
 +
    <td>Multiple unit cell</td>
 +
    <td>Unit cell volume ''V*<sub>c</sub>''</td>
 +
</tr> 
 +
<tr align=center>
 +
<td>''P''</td>
 +
    <td>0</td>
 +
    <td>''V''</td>
 +
    <td>''P''</td>
 +
    <td>'''a*<sub>c</sub>''', '''b*<sub>c</sub>''', '''c*<sub>c</sub>'''</td>
 +
    <td>''V*''</td>
 +
</tr>
 +
<tr align=center>
 +
<td>''A''</td>
 +
    <td>&#189;'''b<sub>c</sub>'''+&#189;'''c<sub>c</sub>'''</td>
 +
    <td>2''V''</td>
 +
    <td>''A''</td>
 +
    <td>'''a*<sub>c</sub>''', 2'''b*<sub>c</sub>''', 2'''c*<sub>c</sub>'''</td>
 +
    <td>&#189;''V*''</td>
 +
</tr>
 +
<tr align=center>
 +
<td>''B''</td>
 +
    <td>&#189;'''c<sub>c</sub>'''+&#189;'''a<sub>c</sub>'''</td>
 +
    <td>2''V''</td>
 +
    <td>''B''</td>
 +
    <td>2'''a*<sub>c</sub>''', '''b*<sub>c</sub>''', 2'''c*<sub>c</sub>'''</td>
 +
    <td>&#189;''V*''</td>
 +
</tr>
 +
<tr align=center>
 +
<td>''C''</td>
 +
    <td>&#189;'''a<sub>c</sub>'''+&#189;'''b<sub>c</sub>'''</td>
 +
    <td>2''V''</td>
 +
    <td>''C''</td>
 +
    <td>2'''a*<sub>c</sub>''', 2'''b*<sub>c</sub>''', '''c*<sub>c</sub>'''</td>
 +
    <td>&#189;''V*''</td>
 +
</tr>
 +
<tr align=center>
 +
<td>''I''</td>
 +
    <td>&#189;'''a<sub>c</sub>'''+ &#189;'''b<sub>c</sub>''' +&#189;'''c<sub>c</sub>'''</td>
 +
    <td>2''V''</td>
 +
    <td>''F''</td>
 +
    <td>2'''a*<sub>c</sub>''', 2'''b*<sub>c</sub>''', 2'''c*<sub>c</sub>'''</td>
 +
    <td>&#189;''V*''</td>
 +
<tr align=center>
 +
<td rowspan=3>''F''</td>
 +
    <td>&#189;'''a<sub>c</sub>'''+ &#189;'''b<sub>c</sub>''' </td>
 +
    <td rowspan=3>4''V''</td>
 +
    <td rowspan=3>''I''</td>
 +
    <td rowspan=3>2'''a*<sub>c</sub>''', 2'''b*<sub>c</sub>''', 2'''c*<sub>c</sub>'''</td>
 +
    <td rowspan=3>&#188;''V*''</td>
 +
</tr>
 +
<tr align=center>
 +
      <td>&#189;'''b<sub>c</sub>'''+ &#189;'''c<sub>c</sub>'''</td>
 +
</tr>
 +
<tr align=center>
 +
      <td>&#189;'''c<sub>c</sub>'''+ &#189;'''a<sub>c</sub>'''</td>
 +
</tr>
 +
<tr align=center>
 +
    <td>''R''</td>
 +
    <td rowspan=2>0</td>
 +
    <td rowspan=2>''V''</td>
 +
    <td rowspan=2>''R''</td>
 +
    <td rowspan=2>'''a*<sub>c</sub>''', '''b*<sub>c</sub>''', '''c*<sub>c</sub>'''</td>
 +
    <td rowspan=2>''V*''</td>
 +
</tr>
 +
<tr align=center>
 +
    <td> (rhombohedral axes)
 +
    </td>
 +
  </tr> 
 +
<tr align=center>
 +
    <td>''R''</td>
 +
    <td rowspan=1>&#8532;'''a*<sub>c</sub>''' + &#8531;'''b*<sub>c</sub>''' + &#8531;'''c*<sub>c</sub>'''</td>
 +
    <td rowspan=2>3''V''</td>
 +
    <td rowspan=2>''R''</td>
 +
    <td rowspan=2>3'''a*<sub>c</sub>''', 3'''b*<sub>c</sub>''', 3'''c*<sub>c</sub>'''</td>
 +
    <td rowspan=2>&#8531;''V*''</td>
 +
</tr>
 +
<tr align=center>
 +
    <td> (hexagonal axes)
 +
    </td>
 +
    <td rowspan=1>&#8531;'''a*<sub>c</sub>''' + &#8532;'''b*<sub>c</sub>''' + &#8532;'''c*<sub>c</sub>'''
 +
    </td>
  
The reciprocal lattice of face-centred lattice ''F'' is a body-centred lattice ''I'' and, reciprocally, the reciprocal lattice of body-centred lattice ''I'' is a face-centred lattice ''F'' (see [http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]). The reciprocal lattice of a ''C'' (or ''A'', ''B'') lattice is also a ''C'' (or ''A'', ''B'') lattice. The reciprocal lattice of an ''R'' lattice is also an ''R'' lattice
+
  </tr> 
 +
 +
</table>
 +
 
 +
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''] (Teaching Pamphlet No. 4 of the International Union of Crystallography).  
  
 
== Diffraction condition in reciprocal space ==
 
== Diffraction condition in reciprocal space ==
  
 +
[[Image:ReciprocalLattice-1.gif|right]]
 
The condition that the waves outgoing from two point scatterers separated by a lattice vector  
 
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, ''K''. This condition is satisfied if  
+
'''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:
('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''', where ''h'', ''k'', ''l'' are integers, namely if the diffraction vector  
+
<center>
('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) is a vector '''0H''' of the reciprocal lattice. A node of the reciprocal lattice is therefore associated to each Bragg reflection on the lattice
+
('''s<sub>h</sub>'''/&#955; - '''s<sub>o</sub>'''/&#955;) = ''h'' '''a*''' + ''k'' '''b*''' + ''l'' '''c*''',
planes of Miller indices (''h/K'', ''k/K'', ''l/K''). It is called the ''hkl'' reflection.  
+
</center>
 +
where ''h'', ''k'', ''l'' are integers, namely the diffraction vector '''OH''' is a vector of the reciprocal lattice (Fig. 1).  
  
==== History ====
+
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 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). 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 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.
(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 to a family of lattice planes.  
 
  
+
[[Image:ReciprocalLattice-2.gif]]
  
 +
Consider the lattice plane passing through lattice point ''Q'' and perpendicular to reciprocal-lattice vector '''OH''' and let &#952; be the angle between the incident, '''s<sub>o</sub>''', or the reflected, '''s<sub>h</sub>''', directions and the lattice plane. It can be seen from the figure that
 +
<center>
 +
''OH''/2 = sin &#952;/&#955;,
 +
</center>
 +
and, since ''OH'' = ''n''/''d'' (''d'' lattice spacing of the family of lattice planes associated with '''OH''') and ''d<sub>hkl</sub>'' = ''d''/''n'':
 +
<center>
 +
2 ''d'' sin &#952; = ''n'' &#955;, or 2 ''d<sub>hkl</sub>'' sin &#952; = &#955;,
 +
</center>
 +
which is [[Bragg's law]]. ''n'' is the order of the reflection.
  
=== See also ===
+
[[Image:ReciprocalLattice-3.gif|right]]
  
[[reciprocal space]]
+
Another way to express the diffraction condition in reciprocal space is to consider a sphere centred at a node ''Q'' of the direct lattice, of radius 1/&#955; and passing through the origin ''O'' of the reciprocal lattice (Fig. 3). If it passes through another node, ''H'', of the reciprocal lattice, Bragg's law is satisfied for the family of direct lattice planes associated with that node and of lattice spacing ''d<sub>hkl</sub>'' = ''n''/''OH'' if ''H'' is the ''n''th node on the row '''OH''' (''n'' = 2 in the example of Fig. 3). This sphere is called the [[Ewald sphere]].
  
[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')
 
  
Section 5.1, ''International Tables of Crystallography, Volume A''
 
  
Section 1.1, ''International Tables of Crystallography, Volume B''
+
== History ==
  
Section 1.1, ''International Tables of Crystallography, Volume C''
+
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
 +
''polar axes'', introduced by Bravais in 1850, which associates the direction of its normal with a family of lattice planes.
  
Section 1.1.2, ''International Tables of Crystallography, Volume D''
+
== See also ==
  
 +
*[[Direct lattice]]
 +
*[[Polar lattice]]
 +
*[[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 of ''International Tables for Crystallography, Volume A,'' 6th edition
 +
*Chapter 1.1 of ''International Tables for Crystallography, Volume B''
 +
*Chapter 1.1 of ''International Tables for Crystallography, Volume C''
 +
*Chapter 1.1.2 of ''International Tables for Crystallography, Volume D''
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Latest revision as of 10:36, 15 December 2017

Réseau réciproque (Fr). Reziprokes Gitter (Ge). Reticolo reciproco (It). 逆格子 (Ja). Red recíproca (Sp).


Definition

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:

OH = r*hkl = h a* + k b* + l c*.

If H is the nth node on the row OH, one has:

OH = n OH1 = n (h1 a* + k1 b* + l1 c*),

where H1 is the first node on the row OH and h1, k1, l1 are relatively prime.

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*hkl = 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/OH1 = n/OH if H is the nth node on the reciprocal lattice row OH. One usually sets dhkl = 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 OH1OP = 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 OH1OP = h1x + k1y + l1z = K. The Miller indices of the family are h1, k1, l1. The subscripts of the Miller indices will be dropped hereafter.

The Miller indices of the family of lattice planes parallel to two direct space vectors, r1 = u1 a + v1 b + w1 c and r2 = u2 a + v2 b + w2 c are proportional to the coordinates in reciprocal space, h, k, l, of the vector product of these two vectors:

h/(v1 w2 - v2 w1) = k/(w1 u2 - w2 u1) = l/(u1 v2 - u2 v1).

The coordinates u, v, w in direct space of the zone axis intersection of two families of lattice planes of Miller indices h1, k1, l1 and h2, k2, l2, respectively, are proportional to the coordinates of the vector product of the reciprocal lattice vectors associated with these two families:

u/(k1 l2 - k2 l1) = v/(l1 h2 - l2 h1) = w/(h1 k2 - h2 k1).

Centred lattices

Direct lattice Reciprocal lattice
Bravais letter Centring vectors Unit-cell volume Vc Bravais letter Multiple unit cell Unit cell volume V*c
P 0 V P a*c, b*c, c*c V*
A ½bccc 2V A a*c, 2b*c, 2c*c ½V*
B ½ccac 2V B 2a*c, b*c, 2c*c ½V*
C ½acbc 2V C 2a*c, 2b*c, c*c ½V*
I ½ac+ ½bccc 2V F 2a*c, 2b*c, 2c*c ½V*
F ½ac+ ½bc 4V I 2a*c, 2b*c, 2c*c ¼V*
½bc+ ½cc
½cc+ ½ac
R 0 V R a*c, b*c, c*c V*
(rhombohedral axes)
R a*c + ⅓b*c + ⅓c*c 3V R 3a*c, 3b*c, 3c*c V*
(hexagonal axes) a*c + ⅔b*c + ⅔c*c

where ac, bc, cc are the basis vectors of the conventional multiple cell and a*c, b*c, c*c 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 The Reciprocal Lattice (Teaching Pamphlet No. 4 of the International Union of Crystallography).

Diffraction condition in reciprocal space

ReciprocalLattice-1.gif

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 (sh/λ - so/λ) ⋅ r, where sh and so 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 = sh/λ - so/λ) is of the form:

(sh/λ - so/λ) = h a* + k b* + l c*,

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 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 sh/λ - so/λ = OH generalizes the Laue equations. It is equivalent to Bragg's law, as can be seen in Fig. 2.

ReciprocalLattice-2.gif

Consider the lattice plane passing through lattice point Q and perpendicular to reciprocal-lattice vector OH and let θ be the angle between the incident, so, or the reflected, sh, directions and the lattice plane. It can be seen from the figure that

OH/2 = sin θ/λ,

and, since OH = n/d (d lattice spacing of the family of lattice planes associated with OH) and dhkl = d/n:

2 d sin θ = n λ, or 2 dhkl sin θ = λ,

which is Bragg's law. n is the order of the reflection.

ReciprocalLattice-3.gif

Another way to express the diffraction condition in reciprocal space is to consider a sphere centred at a node Q of the direct lattice, of radius 1/λ and passing through the origin O of the reciprocal lattice (Fig. 3). If it passes through another node, H, of the reciprocal lattice, Bragg's law is satisfied for the family of direct lattice planes associated with that node and of lattice spacing dhkl = n/OH if H is the nth node on the row OH (n = 2 in the example of Fig. 3). This sphere is called the Ewald sphere.


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 (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

  • Direct lattice
  • Polar lattice
  • Reciprocal space
  • The Reciprocal Lattice (Teaching Pamphlet No. 4 of the International Union of Crystallography)
  • Chapter 1.3.2.5 of International Tables for Crystallography, Volume A, 6th edition
  • Chapter 1.1 of International Tables for Crystallography, Volume B
  • Chapter 1.1 of International Tables for Crystallography, Volume C
  • Chapter 1.1.2 of International Tables for Crystallography, Volume D