Metric tensor

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= 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'').
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=== Other languages ===
 
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Tenseur métrique (''Fr'').
 
== 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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'''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>''.
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In a three-dimensional space with basis vectors '''a''', '''b''', '''c''', the coefficients ''g,,ij,,'' of the metric tensor 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:
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''g,,11,,'' = '''a^2^'''; ''g,,12,,'' = '''a . b'''; ''g,,13,,'' = '''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>
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''g,,21,,'' = '''b . a'''; ''g,,22,,'' = '''b^2^'''; ''g,,23,,'' = '''b . c''';[[BR]]
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''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,,31,,'' = '''c . a'''; ''g,,32,,'' = '''c . b'''; ''g,,33,,'' = '''c^2^''';[[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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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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The  inverse matrix of ''g,,ij,,'', ''g^ij^'', (''g^ik^g,,kj,,'' = &#948;''^k^,,j,,'', Kronecker symbol, = 0 if ''i'' &#8800; ''j'', = 1 if ''i'' = ''j'') relates the ["dual basis"], or ["reciprocal space"] vectors '''e^i^''' to the direct basis vectors '''e,,i,,''' through the relations:
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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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'''e^j^''' = ''g^ij^'' '''e,,j,,'''
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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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In three-dimensional space, the dual basis vectors are identical to the ["reciprocal space"] vectors and the components of ''g^ij^'' are:
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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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  ''g^11^'' = '''a*^2^'''; ''g^12^'' = '''a* . b*'''; ''g^13^'' = '''a* . c*''';[[BR]]
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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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''g^21^'' = '''b* . a*'''; ''g^22^'' = '''b*^2^'''; ''g^23^'' = '''b* . c*''';[[BR]]
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''g^31^'' = '''c* . a*'''; ''g^32^'' = '''c* . b*'''; ''g^33^'' = '''c*^2^''';[[BR]]
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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''.
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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>11</sup>'' = '''a*<sup>2</sup>'''; ''g<sup>12</sup>'' = '''a* . b*'''; ''g<sup>13</sup>'' = '''a* . c*''';<br>
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''g<sup>21</sup>'' = '''b* . a*'''; ''g<sup>22</sup>'' = '''b*<sup>2</sup>'''; ''g<sup>23</sup>'' = '''b* . c*''';<br>
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''g<sup>31</sup>'' = '''c* . a*'''; ''g<sup>32</sup>'' = '''c* . b*'''; ''g<sup>33</sup>'' = '''c*<sup>2</sup>''';
with:
with:
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''g^11^'' =  ''b''^2^''c''^2^ sin^2^ &#945;/ V^2^;
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''g<sup>11</sup>'' =  ''b''<sup>2</sup>''c''<sup>2</sup> sin<sup>2</sup> &#945;/ V<sup>2</sup>;
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''g^22^'' =  ''c''^2^''a''^2^ sin^2^ &#946;/ V^2^;
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''g<sup>22</sup>'' =  ''c''<sup>2</sup>''a''<sup>2</sup> sin<sup>2</sup> &#946;/ V<sup>2</sup>;
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''g^33^'' =  ''a''^2^''b''^2^ sin^2^ &#947;/ V^2^;[[BR]]
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''g<sup>33</sup>'' =  ''a''<sup>2</sup>''b''<sup>2</sup> sin<sup>2</sup> &#947;/ V<sup>2</sup>;
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''g^12^'' = ''g^21^'' = (''abc''^2^/ V^2^)(cos &#945; cos &#946; - cos &#947;);
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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;);
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''g^23^'' = ''g^32^'' = (''a^2^bc''/ V^2^)(cos &#946; cos &#947; - cos &#945;);
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''g^31^'' = ''g^13^'' = (''ab^2^c''/ V^2^)(cos &#947; cos &#945; - cos &#946;).
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where ''V'' is the volume of the unit cell ('''a''', '''b''', '''c''')
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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;);
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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;)
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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:
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'''e',,j,, ''' = ''A,,j,, ^i^'' '''e,,i,, '''; ''x'^j^'' = ''B,,i,, ^j^'' ''x^i^'',
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'''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>'',
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where ''A,,j,, ^i^'' and ''B,,i,, ^j^'' are transformation matrices, transpose of one another. According to their
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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
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definition, the components ''g,,ij,,'' of the metric tensor transform like products of basis vectors:
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definition, the components ''g<sub>ij</sub>,'' of the metric tensor transform like products of basis vectors:
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''g',,kl,,'' = ''A,,k,, ^i^A,,l,, ^j^g,,ij,,''.
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''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.  
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The ["dual basis"] vectors and coordinates transform in the change of basis according to:
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The [[dual basis]] vectors and coordinates transform in the change of basis according to:
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'''e'^j^ ''' = ''B,,i,,^j^'' '''e^i^'''; ''x',,j,,'' = ''A,,j,, ^i^x,,i,,'',
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'''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>'',
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and the components ''g^ij^'' transform like products of dual basis vectors:
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and the components ''g<sup>ij</sup>'' transform like products of dual basis vectors:
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''g'^kl^'' = ''A,,i,,^k^ A,,j^l^ g^ij^''.
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''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.
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The mixed components, ''g^i^,,j,,'' = &#948;^i^,,j,, , are once covariant and once contravariant and are invariant.
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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 ==
== Properties of the metric tensor ==
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* The '''tensor nature''' of the metric tensor is demonstrated by the behaviour of its components in a change of basis. The components ''g,,ij,,'' and ''g^ij^'' are the components of a ''unique'' tensor.
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* 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.
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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,,ij,,'' 's and the ''g^ij^'' 's:
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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:
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''V'' ^2^ = &#916; (''g,,ij,,'') = ''abc''(1 - cos ^2^ &#945; - cos ^2^ &#946; - cos ^2^ &#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;)
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''V*''<sup>2</sup> = &#916; (''g<sup>ij</sup>'') = 1/ ''V'' <sup>2</sup>.
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''V*''<sup>2</sup> = &#916; (''g<sup>ij</sup>'') = 1/ ''V''<sup> 2</sup>.
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* 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:
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* 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:
   
   
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''g,,im,,t^ij..^,,kl..,,'' = ''t^j..^,,klm..,,''
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''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.
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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 No. 10 of the International Union of Crystallography)
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Section 1.1.3 of ''International Tables of Crystallography, Volume B''
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*Chapter 1.1.3 of ''International Tables for Crystallography, Volume B''
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*Chapter 1.1.2 of ''International Tables for Crystallography, Volume D''
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Section 1.1.2 of ''International Tables of Crystallography, Volume D''
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----
 
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[[Category:Fundamental crystallography]]
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[[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