View source for Dual basis

<Font color = "blue">Base duale </font>(''Fr''). <Font color="black"> Base duale </Font>(''It'')

== 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 '''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
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>'',
 
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:

''x<sup> j</sup>'' = ''x<sub>i</sub> g<sup>ij</sup>''

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<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<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<sub>i</sub>'' '''e<sup>i</sup>'''

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:

'''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:

''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<sup>ik</sup>'' and ''g<sub>ij</sub>'' are inverse:

''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.

== Change of basis ==

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<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,
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><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''.

=== 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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[[Category:Fundamental crystallography]]<br>
[[Category:Physical properties of crystals]]<br>