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= Dual basis =
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<font color="orange">أساس مزدوج</font> (''Ar''). <font color = "blue">Base duale</font> (''Fr''). <font color="red">Duale Basis</font> (''Ge''). <font color="black">Base duale</font> (''It''). <font color="purple">双対基底</font> (''Ja''). <font color="green">Base dual</font> (''Sp'').
 
 
=== Other languages ===
 
 
 
Base duale (''Fr'').
 
  
 
== Definition ==
 
== Definition ==
  
The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows:
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The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows.
  
 
Given a basis of ''n'' vectors '''e<sub>i</sub>''' spanning the [[direct space]] ''E<sup>n</sup>'', and a vector
 
Given a basis of ''n'' vectors '''e<sub>i</sub>''' spanning the [[direct space]] ''E<sup>n</sup>'', and a vector
'''x''' = ''x<sup>i</sup>'' '''e<sub>i</sub>''', let us consider the ''n'' quantities defined by the
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'''x''' = ''x<sup> i</sup>'' '''e<sub>i</sub>''', let us consider the ''n'' quantities defined by the
 
scalar products of '''x''' with the basis vectors, '''e<sub>i</sub>''':
 
scalar products of '''x''' with the basis vectors, '''e<sub>i</sub>''':
  
''x<sub>i</sub>'' = '''x''' . '''e<sub>i</sub>''' = ''x<sup>j</sup>'' '''e<sub>j</sub>''' . '''e<sub>i</sub>''' = ''x<sup>j</sup> g<sub>ji</sub>'',
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''x<sub>i</sub>'' = '''x''' . '''e<sub>i</sub>''' = ''x<sup> j</sup>'' '''e<sub>j</sub>''' . '''e<sub>i</sub>''' = ''x<sup> j</sup> g<sub>ji</sub>'',
 
   
 
   
 
where the ''g<sub>ji</sub>'' 's are the doubly covariant components of the [[metric tensor]].
 
where the ''g<sub>ji</sub>'' 's are the doubly covariant components of the [[metric tensor]].
  
By solving these equations in terms of ''x<sup>j</sup>'', one gets:
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By solving these equations in terms of ''x<sup> j</sup>'', one gets
  
''x^j^'' = ''x,,i,, g^ij^''
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''x<sup> j</sup>'' = ''x<sub>i</sub> g<sup>ij</sup>''
  
where the matrix of the  ''g^ij^'' 's is inverse of that of the ''g,,ij,,'' 's (''g^ik^g,,jk,,'' = &#948;^i^,,j,,). The development of vector '''x''' with respect to basis vectors '''e,,i,,''' can now also be written:
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where the matrix of the  ''g<sup>ij</sup>'' 's is inverse of that of the ''g<sub>ij</sub>'' 's (''g<sup>ik</sup>g<sub>jk</sub>'' = &#948;<sup>i</sup><sub>j</sub>). The development of vector '''x''' with respect to basis vectors '''e<sub>i</sub>''' can now also be written
  
'''x''' = ''x^i^'' '''e,,i,,''' = ''x,,i,, g^ij^'' '''e,,j,,'''
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'''x''' = ''x<sup> i</sup>'' '''e<sub>i</sub>''' = ''x<sub>i</sub> g<sup>ij</sup>'' '''e<sub>j</sub>'''.
  
The set of ''n'' vectors '''e^i^''' = ''g^ij^'' '''e,,j,,''' that span the space ''E^n^'' forms a basis since vector '''x''' can be written:
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The set of ''n'' vectors '''e<sup>i</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>''' that span the space ''E<sup>n</sup>'' forms a basis since vector '''x''' can be written
  
'''x''' = ''x,,i,,'' '''e^i^'''
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'''x''' = ''x<sub>i</sub>'' '''e<sup>i</sup>'''.
  
This basis is the ''dual basis'' and the ''n'' quantities ''x,,i,,'' defined above are the
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This basis is the ''dual basis'' and the ''n'' quantities ''x<sub>i</sub>'' defined above are the
 
coordinates of '''x''' with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:
 
coordinates of '''x''' with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:
  
'''e,,i,,''' = ''g,,ij,,'' '''e^j^'''
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'''e<sub>i</sub>''' = ''g<sub>ij</sub>'' '''e<sup>j</sup>'''.
  
The scalar products of the basis vectors of the dual and direct bases are:
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The scalar products of the basis vectors of the dual and direct bases are
  
''g^i^,,j,,'' = '''e^i^''' . '''e,,j,,''' = ''g^ik^'' '''e,,k,,''' . '''e,,j,,''' = ''g^ik^g,,jk,,'' = &#948;^i^,,j,,.
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''g<sup>i</sup><sub>j</sub>'' = '''e<sup>i</sup>''' . '''e<sub>j</sub>''' = ''g<sup>ik</sup>'' '''e<sub>k</sub>''' . '''e<sub>j</sub>''' = ''g<sup>ik</sup>g<sub>jk</sub>'' = &#948;<sup>i</sup><sub>j</sub>.
  
One has therefore, since the matrices ''g^ik^'' and ''g,,ij,,'' are inverse:
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One has therefore, since the matrices ''g<sup>ik</sup>'' and ''g<sub>ij</sub>'' are inverse,
  
''g^i^,,j,,'' = '''e^i^''' . '''e,,j,,''' = &#948;^i^,,j,,.
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''g<sup>i</sup><sub>j</sub>'' = '''e<sup>i</sup>''' . '''e<sub>j</sub>''' = &#948;<sup>i</sup><sub>j</sub>.
  
These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of ["reciprocal space"] are identical.
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These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of [[reciprocal space]] are identical.
  
 
== Change of basis ==
 
== Change of basis ==
  
In a change of basis where the direct basis vectors and coordinates transform like:
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In a change of basis where the direct basis vectors and coordinates transform like
  
'''e'<sub>j</sub>''' = ''A<sub>j</sub><sup>i</sup>'' '''e<sub>i</sub>'''; ''x'<sup>j</sup>'' = ''B<sub>i</sub?<sup>j</sup>'' ''x^i^'',
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'''e'<sub>j</sub>''' = ''A<sub>j</sub><sup>i</sup>'' '''e<sub>i</sub>'''; ''x'<sup>j</sup>'' = ''B<sub>i</sub><sup> j</sup>'' ''x<sup>i</sup>'',
  
where ''A<sub>j</sub><sup>i</sup>'' and ''B<sub>i</sub><sup>j</sup>'' are transformation matrices, transpose of one another,
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where ''A<sub>j</sub><sup>i</sup>'' and ''B<sub>i</sub><sup> j</sup>'' are transformation matrices, transpose of one another,
the dual basis vectors '''e<sup>i</sup>''' and the coordinates ''x<sub>i</sub>'' transform according to:
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the dual basis vectors '''e<sup>i</sup>''' and the coordinates ''x<sub>i</sub>'' transform according to
  
'''e'<sup>j</sup> ''' = ''B<sub>i</sub>^j^'' '''e<sup>i</sup>'''; ''x'<sub>j</sub>'' = ''A<sub>j</sub><sup>i</sup>x<sub>i</sub>''.
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'''e'<sup>j</sup> ''' = ''B<sub>i</sub><sup> j</sup>'' '''e<sup>i</sup>'''; ''x'<sub>j</sub>'' = ''A<sub>j</sub><sup>i</sup>x<sub>i</sub>''.
  
 
The coordinates of a vector in reciprocal space are therefore ''covariant'' and the dual basis vectors (or reciprocal vectors) ''contravariant''.
 
The coordinates of a vector in reciprocal space are therefore ''covariant'' and the dual basis vectors (or reciprocal vectors) ''contravariant''.
  
=== See also ===
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== See also ==
 
 
[[metric tensor]]<br>
 
[[reciprocal space]]<br>
 
 
 
[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ The Reciprocal Lattice]  (Teaching Pamphlet of the ''International Union of Crystallography'')
 
 
 
Section 1.1.2 of ''International Tables of Crystallography, Volume D''
 
 
 
  
----
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*[[Metric tensor]]
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*[[Reciprocal space]]
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*[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/4/ ''The Reciprocal Lattice'']  (Teaching Pamphlet No. 4 of the International Union of Crystallography)
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*Chapter 1.1.2 of ''International Tables for Crystallography, Volume D''
  
 
[[Category:Fundamental crystallography]]<br>
 
[[Category:Fundamental crystallography]]<br>
 
[[Category:Physical properties of crystals]]<br>
 
[[Category:Physical properties of crystals]]<br>

Latest revision as of 13:56, 10 November 2017

أساس مزدوج (Ar). Base duale (Fr). Duale Basis (Ge). Base duale (It). 双対基底 (Ja). Base dual (Sp).

Definition

The dual basis is a basis associated to the basis of a vector space. In three-dimensional space, it is isomorphous to the basis of the reciprocal lattice. It is mathematically defined as follows.

Given a basis of n vectors ei spanning the direct space En, and a vector x = x i ei, let us consider the n quantities defined by the scalar products of x with the basis vectors, ei:

xi = x . ei = x j ej . ei = x j gji,

where the gji 's are the doubly covariant components of the metric tensor.

By solving these equations in terms of x j, one gets

x j = xi gij

where the matrix of the gij 's is inverse of that of the gij 's (gikgjk = δij). The development of vector x with respect to basis vectors ei can now also be written

x = x i ei = xi gij ej.

The set of n vectors ei = gij ej that span the space En forms a basis since vector x can be written

x = xi ei.

This basis is the dual basis and the n quantities xi defined above are the coordinates of x with respect to the dual basis. In a similar way one can express the direct basis vectors in terms of the dual basis vectors:

ei = gij ej.

The scalar products of the basis vectors of the dual and direct bases are

gij = ei . ej = gik ek . ej = gikgjk = δij.

One has therefore, since the matrices gik and gij are inverse,

gij = ei . ej = δij.

These relations show that the dual basis vectors satisfy the definition conditions of the reciprocal vectors. In a three-dimensional space the dual basis and the basis of reciprocal space are identical.

Change of basis

In a change of basis where the direct basis vectors and coordinates transform like

e'j = Aji ei; x'j = Bi j xi,

where Aji and Bi j are transformation matrices, transpose of one another, the dual basis vectors ei and the coordinates xi transform according to

e'j = Bi j ei; x'j = Ajixi.

The coordinates of a vector in reciprocal space are therefore covariant and the dual basis vectors (or reciprocal vectors) contravariant.

See also