# Metric tensor

### From Online Dictionary of Crystallography

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'''x''' . '''y''' = ''x<sup>i</sup>'' '''e<sub>i</sub>''' . ''y<sup>j</sup>'' '''e<sub>j</sub>''' = ''g<sub>ij</sub>'' ''x<sup>i</sup>'' ''y<sup>j</sup>''. | '''x''' . '''y''' = ''x<sup>i</sup>'' '''e<sub>i</sub>''' . ''y<sup>j</sup>'' '''e<sub>j</sub>''' = ''g<sub>ij</sub>'' ''x<sup>i</sup>'' ''y<sup>j</sup>''. | ||

- | In a three-dimensional space with basis vectors '''a''', '''b''', '''c''', the coefficients ''g | + | In a three-dimensional space with basis vectors '''a''', '''b''', '''c''', the coefficients ''g<sub>ij</sub>'' of the metric tensor are: |

- | + | ''g<sub>11</sub>,'' = '''a<sup>2</sup>'''; ''g<sub>12</sub>'' = '''a . b'''; ''g<sub>13</sub>'' = '''a . c''';<br> | |

- | + | ''g<sub>21</sub>'' = '''b . a'''; ''g<sub>22</sub>'' = '''b<sup>2</sup>'''; ''g<sub>23</sub>'' = '''b . c''';<br> | |

- | + | ''g<sub>31</sub>'' = '''c . a'''; ''g<sub>32</sub>'' = '''c . b'''; ''g<sub>33</sub>'' = '''c<sup>2</sup>'''; | |

+ | The inverse matrix of ''g<sub>ij</sub>'', ''g<sup>ij</sup>'', (''g<sup>ik</sup>g<sub>kj</sub>'' = δ''<sup>k</sup><sub>j</sub>'', Kronecker symbol, = 0 if ''i'' ≠ ''j'', = 1 if ''i'' = ''j'') relates the [[dual basis]], or [[reciprocal space]] vectors '''e<sup>i</sup>''' to the direct basis vectors '''e<sub>i</sub>''' through the relations: | ||

- | + | '''e<sup>j</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>''' | |

- | + | In three-dimensional space, the dual basis vectors are identical to the [[reciprocal space]] vectors and the components of ''g<sup>ij</sup>'' are: | |

- | + | ||

- | In three-dimensional space, the dual basis vectors are identical to the [ | + | |

''g^11^'' = '''a*^2^'''; ''g^12^'' = '''a* . b*'''; ''g^13^'' = '''a* . c*''';[[BR]] | ''g^11^'' = '''a*^2^'''; ''g^12^'' = '''a* . b*'''; ''g^13^'' = '''a* . c*''';[[BR]] |

## Revision as of 14:28, 25 January 2006

## Contents |

# Metric tensor

### Other languages

Tenseur métrique (*Fr*).

## Definition

Given a basis **e _{i}** of a

*Euclidean space*,

*E*, the metric tensor is a rank 2 tensor the components of which are:

^{n}*g _{ij}* =

**e**.

_{i}**e**=

_{j}**e**.

_{j}**e**=

_{i}*g*.

_{ji}It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, **x** = *x ^{i}*

**e**and

_{i}**y**=

*y*

^{j}**e**is written:

_{j}**x** . **y** = *x ^{i}*

**e**.

_{i}*y*

^{j}**e**=

_{j}*g*

_{ij}*x*

^{i}*y*.

^{j}In a three-dimensional space with basis vectors **a**, **b**, **c**, the coefficients *g _{ij}* of the metric tensor are:

*g _{11},* =

**a**;

^{2}*g*=

_{12}**a . b**;

*g*=

_{13}**a . c**;

*g*=

_{21}**b . a**;

*g*=

_{22}**b**;

^{2}*g*=

_{23}**b . c**;

*g*=

_{31}**c . a**;

*g*=

_{32}**c . b**;

*g*=

_{33}**c**;

^{2}The inverse matrix of *g _{ij}*,

*g*, (

^{ij}*g*= δ

^{ik}g_{kj}*, Kronecker symbol, = 0 if*

^{k}_{j}*i*≠

*j*, = 1 if

*i*=

*j*) relates the dual basis, or reciprocal space vectors

**e**to the direct basis vectors

^{i}**e**through the relations:

_{i}**e ^{j}** =

*g*

^{ij}**e**

_{j}In three-dimensional space, the dual basis vectors are identical to the reciprocal space vectors and the components of *g ^{ij}* are:

g^11^=a*^2^;g^12^=a* . b*;g^13^=a* . c*;BRg^21^=b* . a*;g^22^=b*^2^;g^23^=b* . c*;BRg^31^=c* . a*;g^32^=c* . b*;g^33^=c*^2^;BR

with:

g^11^=b^2^c^2^ sin^2^ α/ V^2^;g^22^=c^2^a^2^ sin^2^ β/ V^2^;g^33^=a^2^b^2^ sin^2^ γ/ V^2^;BR

g^12^=g^21^= (abc^2^/ V^2^)(cos α cos β - cos γ);g^23^=g^32^= (a^2^bc/ V^2^)(cos β cos γ - cos α);g^31^=g^13^= (ab^2^c/ V^2^)(cos γ cos α - cos β).

where *V* is the volume of the unit cell (**a**, **b**, **c**)

## Change of basis

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

e',,j,,=A,,j,, ^i^e,,i,,;x'^j^=B,,i,, ^j^x^i^,

where *A,,j,, ^i^* and *B,,i,, ^j^* are transformation matrices, transpose of one another. According to their
definition, the components *g,,ij,,* of the metric tensor transform like products of basis vectors:

g',,kl,,=A,,k,, ^i^A,,l,, ^j^g,,ij,,.

They are the doubly covariant components of the metric tensor.

The ["dual basis"] vectors and coordinates transform in the change of basis according to:

e'^j^=B,,i,,^j^e^i^;x',,j,,=A,,j,, ^i^x,,i,,,

and the components *g^ij^* transform like products of dual basis vectors:

g'^kl^=A,,i,,^k^ A,,j^l^ g^ij^.

They are the doubly contravariant components of the metric tensor.

The mixed components, *g^i^,,j,,* = δ^i^,,j,, , are once covariant and once contravariant and are invariant.

## Properties of the metric tensor

* Thetensor natureof the metric tensor is demonstrated by the behaviour of its components in a change of basis. The componentsg,,ij,,andg^ij^are the components of auniquetensor.

* Thesquares of the volumesVandV*of the direct space and reciprocal space unit cells are respectively equal to the determinants of theg,,ij,,'s and theg^ij^'s:

V^2^ = Δ (g,,ij,,) =abc(1 - cos ^2^ α - cos ^2^ β - cos ^2^ γ + 2 cos α cos α cos α)

*V**^{2} = Δ (*g ^{ij}*) = 1/

*V*

^{2}.

* One changes thevariance of a tensorby taking the contracted tensor product of the tensor by the suitable form of the metric tensor. For instance:g,,im,,t^ij..^,,kl..,,=t^j..^,,klm..,,

Multiplying by the doubly covariant form of the metric tensor increases the covariance by one, multiplying by the doubly contravariant form increases the contravariance by one.

## See also

Section 1.1.3 of *International Tables of Crystallography, Volume B*

Section 1.1.2 of *International Tables of Crystallography, Volume D*