Difference between revisions of "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''). |
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| − | === | ||
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| − | Base | ||
== 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 | + | 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 | + | '''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>'', | + | ''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 | + | By solving these equations in terms of ''x<sup> j</sup>'', one gets |
| − | ''x | + | ''x<sup> j</sup>'' = ''x<sub>i</sub> g<sup>ij</sup>'' |
| − | where the matrix of the ''g | + | 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>'' = δ<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 | + | 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 | + | 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<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 | + | 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>'' = δ<sup>i</sup><sub>j</sub>. | |
| − | One has therefore, since the matrices ''g | + | 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>''' = δ<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 [ | + | 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 | + | 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, | + | 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 | + | 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''. | 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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| − | - | + | *[[Metric tensor]] |
| + | *[[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 1.1.2 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
- Metric tensor
- Reciprocal space
- The Reciprocal Lattice (Teaching Pamphlet No. 4 of the International Union of Crystallography)
- Chapter 1.1.2 of International Tables for Crystallography, Volume D