# Difference between revisions of "Metric tensor"

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− | = | + | <font color="blue">Tenseur métrique</font> (''Fr''). <font color="red">Metrischer Tensor</font> (''Ge''). <font color="black">Tensore metrico</font> (''It''). <font color="purple">計量テンソル</font> (''Ja''). <font color="brown">Метрический тензор</font> (''Ru''). <font color="green">Tensor métrico</font> (''Sp''). |

− | |||

− | |||

− | |||

== Definition == | == Definition == | ||

− | + | A metric tensor is used to measure distances in a space. In crystallography the spaces considered are vector spaces with | |

− | + | ''Euclidean'' metrics, <i>i.e.</i> ones for which the rules of Euclidean geometry apply. In that case, | |

+ | given a basis '''e<sub>i</sub>''' of a ''Euclidean space'', ''E<sup>n</sup>'', the metric tensor is a rank 2 tensor the components of | ||

which are: | which are: | ||

Line 16: | Line 14: | ||

'''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>'''. | ||

+ | |||

+ | Because the metric tensor is symmetric, ''g<sub>12</sub>'' = ''g<sub>21</sub>'', ''g<sub>13</sub>'' = ''g<sub>31</sub>'', and ''g<sub>13</sub>'' = ''g<sub>31</sub>''. Thus there are only six unique elements, often presented as | ||

− | + | ''g<sub>11</sub>'' ''g<sub>22</sub>'' ''g<sub>33</sub>'' <br> | |

− | + | ''g<sub>23</sub>'' ''g<sub>13</sub>'' ''g<sub>12</sub>'' <br> | |

− | ''g | ||

+ | or, multiplying the second row by 2, as a so-called G<sup>6</sup> ("G" for Gruber) vector | ||

− | + | ( '''a<sup>2</sup>''', '''b<sup>2</sup>''', '''c<sup>2</sup>''', 2'''b . c''', 2'''a . c''', 2'''a . b''' ) | |

− | '''e | + | The inverse matrix of ''g<sub>ij</sub>'', ''g<sup>ij</sup>'', 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>. | |

− | + | Note that ''g<sup>ik</sup>g<sub>kj</sub>'' = δ''<sup>k</sup><sub>j</sub>'', where δ''<sup>k</sup><sub>j</sub>'' is the Kronecker symbol, equal to 0 if ''i'' ≠ ''j'', and equal to 1 if ''i'' = ''j''. | |

− | + | ||

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

+ | |||

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

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

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

with: | with: | ||

− | + | ''g<sup>11</sup>'' = ''b''<sup>2</sup>''c''<sup>2</sup> sin<sup>2</sup> α/ V<sup>2</sup>; | |

− | + | ''g<sup>22</sup>'' = ''c''<sup>2</sup>''a''<sup>2</sup> sin<sup>2</sup> β/ V<sup>2</sup>; | |

− | + | ''g<sup>33</sup>'' = ''a''<sup>2</sup>''b''<sup>2</sup> sin<sup>2</sup> γ/ V<sup>2</sup>; | |

− | + | ''g<sup>12</sup>'' = ''g<sup>21</sup>'' = (''abc''<sup>2</sup>/ V<sup>2</sup>)(cos α cos β - cos γ); | |

− | |||

− | |||

− | where ''V'' is the volume of the unit cell ('''a''', '''b''', '''c''') | + | ''g<sup>23</sup>'' = ''g<sup>32</sup>'' = (''a<sup>2</sup>bc''/ V<sup>2</sup>)(cos β cos γ - cos α); |

+ | |||

+ | ''g<sup>31</sup>'' = ''g<sup>13</sup>'' = (''ab<sup>2</sup>c''/ V<sup>2</sup>)(cos γ cos α - cos β) | ||

+ | |||

+ | where ''V'' is the volume of the unit cell ('''a''', '''b''', '''c'''). | ||

== Change of basis == | == Change of basis == | ||

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In a change of basis the direct basis vectors and coordinates transform like: | In a change of basis 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 | + | where ''A<sub>j</sub><sup> i</sup>'' and ''B<sub>i</sub><sup> j</sup>'' are transformation matrices, transpose of one another. According to their |

− | definition, the components ''g | + | definition, the components ''g<sub>ij</sub>,'' of the metric tensor transform like products of basis vectors: |

− | + | ''g'<sub>kl</sub>'' = ''A<sub>k</sub><sup>i</sup>A<sub>l</sub><sup>j</sup>g<sub>ij</sub>''. | |

They are the doubly covariant components of the metric tensor. | They are the doubly covariant components of the metric tensor. | ||

− | The [ | + | The [[dual basis]] vectors and coordinates transform in the change of basis 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>'', | |

− | and the components ''g | + | and the components ''g<sup>ij</sup>'' transform like products of dual basis vectors: |

− | + | ''g'<sup>kl</sup>'' = ''A<sub>i</sub><sup>k</sup> A<sub>j</sub><sup>l</sup> g<sup>ij</sup>''. | |

They are the doubly contravariant components of the metric tensor. | They are the doubly contravariant components of the metric tensor. | ||

− | The mixed components, ''g | + | The mixed components, ''g<sup>i</sup><sub>j</sub>'' = δ<sup>i</sup><sub>j</sub>, are once covariant and once contravariant and are invariant. |

== Properties of the metric tensor == | == Properties of the metric tensor == | ||

− | + | * The '''tensor nature''' of the metric tensor is demonstrated by the behaviour of its components in a change of basis. The components ''g<sub>ij</sub>'' and ''g<sup>ij</sup>'' are the components of a ''unique'' tensor. | |

− | + | * The '''squares of the volumes''' ''V'' and ''V*'' of the direct space and reciprocal space unit cells are respectively equal to the determinants of the ''g<sub>ij</sub>'' 's and the ''g<sup>ij</sup>'' 's: | |

− | + | ''V''<sup> 2</sup> = Δ (''g<sub>ij</sub>'') = ''abc''(1 - cos <sup>2</sup> α - cos <sup>2</sup> β - cos<sup>2</sup> γ + 2 cos α cos β cos γ) | |

− | ''V*''<sup>2</sup> = Δ (''g<sup>ij</sup>'') = 1/ ''V'' <sup>2</sup>. | + | ''V*''<sup>2</sup> = Δ (''g<sup>ij</sup>'') = 1/ ''V''<sup> 2</sup>. |

− | + | * One changes the '''variance of a tensor''' by taking the contracted tensor product of the tensor by the suitable form of the metric tensor. For instance: | |

− | + | ''g<sub>im</sub>t<sup> ij..</sup><sub>kl..</sub>'' = ''t<sup> j..</sup><sub>klm..</sub>'' | |

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. | 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 == | == See also == | ||

− | [[ | + | *[[Dual basis]] |

− | [[ | + | *[[Reciprocal space]] |

− | + | *[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/10/ ''Metric Tensor and Symmetry Operations in Crystallography''] (Teaching Pamphlet No. 10 of the International Union of Crystallography) | |

− | + | *Chapter 1.1.3 of ''International Tables for Crystallography, Volume B'' | |

− | + | *Chapter 1.1.2 of ''International Tables for Crystallography, Volume D'' | |

− | |||

− | |||

− | [[Category:Fundamental crystallography]] | + | [[Category:Fundamental crystallography]]<br> |

[[Category:Physical properties of crystals]] | [[Category:Physical properties of crystals]] |

## Latest revision as of 09:21, 11 December 2017

Tenseur métrique (*Fr*). Metrischer Tensor (*Ge*). Tensore metrico (*It*). 計量テンソル (*Ja*). Метрический тензор (*Ru*). Tensor métrico (*Sp*).

## Definition

A metric tensor is used to measure distances in a space. In crystallography the spaces considered are vector spaces with
*Euclidean* metrics, *i.e.* ones for which the rules of Euclidean geometry apply. In that case,
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}Because the metric tensor is symmetric, *g _{12}* =

*g*,

_{21}*g*=

_{13}*g*, and

_{31}*g*=

_{13}*g*. Thus there are only six unique elements, often presented as

_{31}*g _{11}*

*g*

_{22}*g*

_{33}*g*

_{23}*g*

_{13}*g*

_{12}or, multiplying the second row by 2, as a so-called G^{6} ("G" for Gruber) vector

( **a ^{2}**,

**b**,

^{2}**c**, 2

^{2}**b . c**, 2

**a . c**, 2

**a . b**)

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

*g*, relates the dual basis, or reciprocal space vectors

^{ij}**e**

^{i}to the direct basis vectors

**e**

_{i}, through the relations

**e**^{j} = *g ^{ij}*

**e**

_{j}.

Note that *g ^{ik}g_{kj}* = δ

*, where δ*

^{k}_{j}*is the Kronecker symbol, equal to 0 if*

^{k}_{j}*i*≠

*j*, and equal to 1 if

*i*=

*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***;

*g*=

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

*g*=

^{22}**b***;

^{2}*g*=

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

*g*=

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

*g*=

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

*g*=

^{33}**c***;

^{2}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};

*g ^{12}* =

*g*= (

^{21}*abc*

^{2}/ V

^{2})(cos α cos β - cos γ);

*g ^{23}* =

*g*= (

^{32}*a*/ V

^{2}bc^{2})(cos β cos γ - cos α);

*g ^{31}* =

*g*= (

^{13}*ab*/ V

^{2}c^{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*are transformation matrices, transpose of one another. According to their definition, the components

_{i}^{ j}*g*of the metric tensor transform like products of basis vectors:

_{ij},*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

- The
**tensor nature**of the metric tensor is demonstrated by the behaviour of its components in a change of basis. The components*g*and_{ij}*g*are the components of a^{ij}*unique*tensor.

- The
**squares of the volumes***V*and*V**of the direct space and reciprocal space unit cells are respectively equal to the determinants of the*g*'s and the_{ij}*g*'s:^{ij}

*V*^{ 2} = Δ (*g _{ij}*) =

*abc*(1 - cos

^{2}α - cos

^{2}β - cos

^{2}γ + 2 cos α cos β cos γ)

*V**^{2} = Δ (*g ^{ij}*) = 1/

*V*

^{ 2}.

- One changes the
**variance of a tensor**by 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

- Dual basis
- Reciprocal space
*Metric Tensor and Symmetry Operations in Crystallography*(Teaching Pamphlet No. 10 of the International Union of Crystallography)- Chapter 1.1.3 of
*International Tables for Crystallography, Volume B* - Chapter 1.1.2 of
*International Tables for Crystallography, Volume D*