# Difference between revisions of "Metric tensor"

### From Online Dictionary of Crystallography

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m (Add explanation of what a metric tensor is making it clear the definition is for a restricted case; add common crystallographic presentations as 2 rows of 3 or as a G6 vector. remove stray comma.) |
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== 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 14: | Line 15: | ||

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

+ | |||

+ | 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''' ) | ||

+ | |||

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: | 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: |

## Revision as of 21:21, 11 August 2014

Tenseur métrique (*Fr*); Tensore metrico (*It*); 計量テンソル (*Ja*).

## 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*, (

^{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***;

*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 α cos α)

*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 of the
*International Union of Crystallography*) - Section 1.1.3 of
*International Tables of Crystallography, Volume B* - Section 1.1.2 of
*International Tables of Crystallography, Volume D*