Metric tensor

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<Font color="blue">Tenseur métrique </Font>(''Fr''). <Font color="black"> Tensore metrico </Font>(''It'')
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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'').
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== Definition ==
== Definition ==
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A metric tensor is used to measure distances in a space.  In crystallography the spaces considered are vector spaces with
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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  
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''Euclidean'' metrics,  <i>i.e.</i> ones for which the rules of Euclidean geometry apply.  In that case,
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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:
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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:
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''g<sub>11</sub>,'' = '''a<sup>2</sup>'''; ''g<sub>12</sub>'' = '''a . b'''; ''g<sub>13</sub>'' = '''a . c''';<br>
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''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>
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''g<sub>31</sub>'' = '''c . a'''; ''g<sub>32</sub>'' = '''c . b'''; ''g<sub>33</sub>'' = '''c<sup>2</sup>''';
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''g<sub>31</sub>'' = '''c . a'''; ''g<sub>32</sub>'' = '''c . b'''; ''g<sub>33</sub>'' = '''c<sup>2</sup>'''.
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The  inverse matrix of ''g<sub>ij</sub>'', ''g<sup>ij</sup>'', (''g<sup>ik</sup>g<sub>kj</sub>'' = &#948;''<sup>k</sup><sub>j</sub>'', Kronecker symbol, = 0 if ''i'' &#8800; ''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:
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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
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'''e<sup>j</sup>''' = ''g<sup>ij</sup>'' '''e<sub>j</sub>'''
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''g<sub>11</sub>''  ''g<sub>22</sub>''  ''g<sub>33</sub>'' <br>
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''g<sub>23</sub>''  ''g<sub>13</sub>''  ''g<sub>12</sub>'' <br>
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or, multiplying the second row by 2,  as a so-called G<sup>6</sup> ("G" for Gruber) vector
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( '''a<sup>2</sup>''', '''b<sup>2</sup>''', '''c<sup>2</sup>''', 2'''b . c''', 2'''a . c''', 2'''a . b''' )
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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
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'''e'''<sup>''j''</sup> = ''g<sup>ij</sup>'' '''e'''<sub>''j''</sub>.
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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:
In three-dimensional space, the dual basis vectors are identical to the [[reciprocal space]] vectors and the components of ''g<sup>ij</sup>'' are:
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''g<sup>12</sup>'' = ''g<sup>21</sup>'' = (''abc''<sup>2</sup>/ V<sup>2</sup>)(cos &#945; cos &#946; - cos &#947;);
''g<sup>12</sup>'' = ''g<sup>21</sup>'' = (''abc''<sup>2</sup>/ V<sup>2</sup>)(cos &#945; cos &#946; - cos &#947;);
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''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>23</sup>'' = ''g<sup>32</sup>'' = (''a<sup>2</sup>bc''/ V<sup>2</sup>)(cos &#946; cos &#947; - cos &#945;);
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''g<sup>31</sup>'' = ''g<sup>13</sup>'' = (''ab<sup>2</sup>c''/ V<sup>2</sup>)(cos &#947; cos &#945; - cos &#946;)
''g<sup>31</sup>'' = ''g<sup>13</sup>'' = (''ab<sup>2</sup>c''/ V<sup>2</sup>)(cos &#947; cos &#945; - cos &#946;)
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* 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:
* 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:
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''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 &#945; cos &#945;)
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''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>.
''V*''<sup>2</sup> = &#916; (''g<sup>ij</sup>'') = 1/ ''V''<sup> 2</sup>.
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== See also ==
== See also ==
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[[dual basis]]<br>
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*[[Dual basis]]
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[[reciprocal space]]<br>
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*[[Reciprocal space]]
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[http://www.iucr.org/iucr-top/comm/cteach/pamphlets/10/ Metric Tensor and Symmetry Operations in Crystallography]  (Teaching Pamphlet of the ''International Union of Crystallography'')<br>
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*[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)
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Section 1.1.3 of ''International Tables of Crystallography, Volume B''<br>
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*Chapter 1.1.3 of ''International Tables for Crystallography, Volume B''
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Section 1.1.2 of ''International Tables of Crystallography, Volume D''<br>
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*Chapter 1.1.2 of ''International Tables for Crystallography, Volume D''
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[[Category:Fundamental crystallography]]<br>
[[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).


Contents

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 ei of a Euclidean space, En, the metric tensor is a rank 2 tensor the components of which are:

gij = ei . ej = ej.ei = gji.

It is a symmetrical tensor. Using the metric tensor, the scalar product of two vectors, x = xi ei and y = yj ej is written:

x . y = xi ei . yj ej = gij xi yj.

In a three-dimensional space with basis vectors a, b, c, the coefficients gij of the metric tensor are:

g11 = a2; g12 = a . b; g13 = a . c;
g21 = b . a; g22 = b2; g23 = b . c;
g31 = c . a; g32 = c . b; g33 = c2.

Because the metric tensor is symmetric, g12 = g21, g13 = g31, and g13 = g31. Thus there are only six unique elements, often presented as

g11 g22 g33
g23 g13 g12

or, multiplying the second row by 2, as a so-called G6 ("G" for Gruber) vector

( a2, b2, c2, 2b . c, 2a . c, 2a . b )

The inverse matrix of gij, gij, relates the dual basis, or reciprocal space vectors ei to the direct basis vectors ei, through the relations

ej = gij ej.

Note that gikgkj = δkj, where δkj is the Kronecker symbol, equal to 0 if ij, 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 gij are:

g11 = a*2; g12 = a* . b*; g13 = a* . c*;
g21 = b* . a*; g22 = b*2; g23 = b* . c*;
g31 = c* . a*; g32 = c* . b*; g33 = c*2;

with:

g11 = b2c2 sin2 α/ V2; g22 = c2a2 sin2 β/ V2; g33 = a2b2 sin2 γ/ V2;

g12 = g21 = (abc2/ V2)(cos α cos β - cos γ);

g23 = g32 = (a2bc/ V2)(cos β cos γ - cos α);

g31 = g13 = (ab2c/ V2)(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 = Aj i ei; x'j = Bi j x i,

where Aj i and Bi j are transformation matrices, transpose of one another. According to their definition, the components gij, of the metric tensor transform like products of basis vectors:

g'kl = AkiAljgij.

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 = Bi j ei; x'j = Aj ixi,

and the components gij transform like products of dual basis vectors:

g'kl = Aik Ajl gij.

They are the doubly contravariant components of the metric tensor.

The mixed components, gij = δij, 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 gij and gij 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 gij 's and the gij 's:

V 2 = Δ (gij) = abc(1 - cos 2 α - cos 2 β - cos2 γ + 2 cos α cos β cos γ)

V*2 = Δ (gij) = 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:

gimt 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