# Metric tensor

(Difference between revisions)
 Revision as of 14:23, 12 January 2007 (view source)← Older edit Latest revision as of 09:21, 11 December 2017 (view source)m (Tidied translations.) (7 intermediate revisions not shown) Line 1: Line 1: - Tenseur métrique (''Fr''). Tensore metrico (''It'') + Tenseur métrique (''Fr''). Metrischer Tensor (''Ge'').  Tensore metrico (''It''). 計量テンソル (''Ja''). Метрический тензор (''Ru''). Tensor métrico (''Sp''). - == Definition == - Given a basis '''ei''' of a ''Euclidean space'', ''En'', the metric tensor is a rank 2 tensor the components of + == 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: which are: Line 14: Line 16: In a three-dimensional space with basis vectors '''a''', '''b''', '''c''', the coefficients ''gij'' of the metric tensor are: 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''';
+ ''g11'' = '''a2'''; ''g12'' = '''a . b'''; ''g13'' = '''a . c''';
''g21'' = '''b . a'''; ''g22'' = '''b2'''; ''g23'' = '''b . c''';
''g21'' = '''b . a'''; ''g22'' = '''b2'''; ''g23'' = '''b . c''';
- ''g31'' = '''c . a'''; ''g32'' = '''c . b'''; ''g33'' = '''c2'''; + ''g31'' = '''c . a'''; ''g32'' = '''c . b'''; ''g33'' = '''c2'''. - The  inverse matrix of ''gij'', ''gij'', (''gikgkj'' = δ''kj'', Kronecker symbol, = 0 if ''i'' ≠ ''j'', = 1 if ''i'' = ''j'') relates the [[dual basis]], or [[reciprocal space]] vectors '''ei''' to the direct basis vectors '''ei''' through the relations: + Because the metric tensor is symmetric, ''g12'' = ''g21'', ''g13'' = ''g31'', and  ''g13'' = ''g31''.  Thus there are only six unique elements, often presented as - '''ej''' = ''gij'' '''ej''' + ''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''', 2'''b . c''', 2'''a . c''', 2'''a . b''' ) + + The  inverse matrix of ''gij'', ''gij'', relates the [[dual basis]], or [[reciprocal space]] vectors '''e'''''i'' to the direct basis vectors '''e'''''i'', through the relations + + '''e'''''j'' = ''gij'' '''e'''''j''. + + Note that ''gikgkj'' = δ''kj'', where δ''kj'' 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 ''gij'' are: In three-dimensional space, the dual basis vectors are identical to the [[reciprocal space]] vectors and the components of ''gij'' are: Line 35: Line 48: ''g12'' = ''g21'' = (''abc''2/ V2)(cos α cos β - cos γ); ''g12'' = ''g21'' = (''abc''2/ V2)(cos α cos β - cos γ); + ''g23'' = ''g32'' = (''a2bc''/ V2)(cos β cos γ - cos α); ''g23'' = ''g32'' = (''a2bc''/ V2)(cos β cos γ - cos α); + ''g31'' = ''g13'' = (''ab2c''/ V2)(cos γ cos α - cos β) ''g31'' = ''g13'' = (''ab2c''/ V2)(cos γ cos α - cos β) Line 71: Line 86: * 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: * 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 α cos α) + ''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. ''V*''2 = Δ (''gij'') = 1/ ''V'' 2. Line 83: Line 98: == See also == == See also == - [[dual basis]]
+ *[[Dual basis]] - [[reciprocal space]]
+ *[[Reciprocal space]] - [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'')
+ *[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) - Section 1.1.3 of ''International Tables of Crystallography, Volume B''
+ *Chapter 1.1.3 of ''International Tables for Crystallography, Volume B'' - Section 1.1.2 of ''International Tables of Crystallography, Volume D''
+ *Chapter 1.1.2 of ''International Tables for Crystallography, Volume D'' - ---- [[Category:Fundamental crystallography]]
[[Category:Fundamental crystallography]]
[[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 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.