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 ==
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:

''g<sub>ij</sub>'' = '''e<sub>i</sub>''' . '''e<sub>j</sub>''' = '''e<sub>j</sub>'''.'''e<sub>i</sub>''' = ''g<sub>ji</sub>''.

It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, '''x''' = ''x<sup>i</sup>'' '''e<sub>i</sub>''' and '''y''' = ''y<sup>j</sup>'' '''e<sub>j</sub>''' is written:

'''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<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>

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>'', 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>'' = &#948;''<sup>k</sup><sub>j</sub>'', where &#948;''<sup>k</sup><sub>j</sub>'' is the Kronecker symbol, equal to 0 if ''i'' &#8800; ''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:

''g<sup>11</sup>'' =  ''b''<sup>2</sup>''c''<sup>2</sup> sin<sup>2</sup> &#945;/ V<sup>2</sup>;
''g<sup>22</sup>'' =  ''c''<sup>2</sup>''a''<sup>2</sup> sin<sup>2</sup> &#946;/ V<sup>2</sup>;
''g<sup>33</sup>'' =  ''a''<sup>2</sup>''b''<sup>2</sup> sin<sup>2</sup> &#947;/ V<sup>2</sup>;

''g<sup>12</sup>'' = ''g<sup>21</sup>'' = (''abc''<sup>2</sup>/ V<sup>2</sup>)(cos &#945; cos &#946; - cos &#947;);

''g<sup>23</sup>'' = ''g<sup>32</sup>'' = (''a<sup>2</sup>bc''/ V<sup>2</sup>)(cos &#946; cos &#947; - cos &#945;);

''g<sup>31</sup>'' = ''g<sup>13</sup>'' = (''ab<sup>2</sup>c''/ V<sup>2</sup>)(cos &#947; cos &#945; - cos &#946;)

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'<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. According to their
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. 

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<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.

The mixed components, ''g<sup>i</sup><sub>j</sub>'' = &#948;<sup>i</sup><sub>j</sub>, 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<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> = &#916; (''g<sub>ij</sub>'') = ''abc''(1 - cos <sup>2</sup> &#945; - cos <sup>2</sup> &#946; - cos<sup>2</sup> &#947; + 2 cos &#945; cos &#946; cos &#947;)

''V*''<sup>2</sup> = &#916; (''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.

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

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